Adapter configured to maintain a surgical tool in a predetermined pose when disconnected from drive unit

ABSTRACT

A surgical tool configured to selectively connect to a drive unit includes an elongate shaft, an end effector, an adapter, and first, second, third, and fourth cables. The end effector includes a first and second jaws. The cables extend through the elongate shaft with a distal portion of the first and fourth cables secured to a first side of the first and second jaws, respectively, and a distal portion of the second and third cables secured to a second side of the first and second jaws, respectively. The adapter is configured to selectively connect to the drive unit and includes a differential drive mechanism configured to manipulate proximal portions of the cables which manipulate the end effector in pitch, yaw, and jaw DOFs. Each of the cables are biased proximally and configured to maintain the end effector in a desired pose when the tool is disconnected from the drive unit.

BACKGROUND

Robotic surgical systems such as teleoperative systems are used toperform minimally invasive surgical procedures that offer many benefitsover traditional open surgery techniques, including less pain, shorterhospital stays, quicker return to normal activities, minimal scarring,reduced recovery time, and less injury to tissue.

Robotic surgical systems can have a number of robotic arms that moveattached instruments or tools, such as an image capturing device, astapler, an electrosurgical instrument, etc., in response to movement ofinput devices by a surgeon viewing images captured by the imagecapturing device of a surgical site. During a surgical procedure, eachof the tools is inserted through an opening, either natural or anincision, into the patient and positioned to manipulate tissue at asurgical site. The openings are placed about the patient's body so thatthe surgical instruments may be used to cooperatively perform thesurgical procedure and the image capturing device may view the surgicalsite.

During the surgical procedure, the tools can include end effectors thatare controlled by one or more open loop cables. The end effector can bemanipulated by controlling the tension in the cables.

There is a continuing need for improved methods for controlling thetension in the cables to manipulate the end effector.

SUMMARY

In an aspect of the present disclosure, a method of controlling an endeffector of a surgical robot includes receiving a desired pose,generating a motor torque for each motor, transmitting the motor torquesfor each motor, generating a null torque for each motor, generating adesired torque for each motor, and transmitting the desired torques toan instrument drive unit (IDU) such that the IDU moves the end effectorto the desired pose. A primary controller receives the desired pose ofthe end effector in three degrees-of-freedom (DOF). The primarycontroller generates the motor torque for each motor of the IDU inresponse to receiving the desired pose. The primary controller transmitsthe motor torques which are received in a secondary controller. Thesecondary controller generates a null torque for each motor of the IDUto maintain tension in cables of a differential drive mechanism of theIDU. The desired torques are generated for each motor of the IDU whichtorques include a sum of the motor torques and the null torques for eachmotor.

In aspects, generating the null torque for each motor of the IDU alsoincludes generating a clamping force between jaws of the end effector.Generating the clamping force may include modifying the desired posesuch that a jaw angle between the jaws of the end effector is negative.The method may include verifying a position of the jaws of the endeffector is less than a clamping threshold before generating theclamping force. The method may include releasing the clamp force whenthe jaws have a position greater than a releasing threshold. Thereleasing threshold may be greater than the clamping threshold.

In some aspects, generating the null torque for each motor includes thesecondary controller receiving a sensed torque from the IDU. The sensedtorque from the IDU may affect the null torque for each motor of theIDU. Generating the null torque for each motor may include adjusting thenull torque for each motor in response to a sensed torque of therespective motor. Adjusting the null torque for each motor may includeapplying a gain to the motor torque of each motor. Generating the nulltorque for each motor may include adjusting the null torque for a pullermotor for each pair of motors of the IDU in response to the sensedtorques.

In certain aspects, generating a desired torque for each motor includesa tertiary controller receiving the motor torques and the null torquesand combining the motor torques and the null torques into desiredtorques including the sum of the motor and null torques. The tertiarycontroller may transmit the desired torques to the IDU. Combining themotor torques and the null torques may include receiving sensed torquesfrom the IDU and applying a gain to the sum of the motor and nulltorques to determine the desired torques such that the sensed torquesapproach the sum of the motor and null torques.

In particular aspects, the secondary controller generates the nulltorque for each motor in response to receiving a motor position for eachmotor of the IDU. The motor position received by the secondarycontroller may be in a joint space. A converter positioned between theIDU and the secondary controller converting a motor position from amotor space to the joint space.

In aspects, generating the motor torque for each motor includescalculating the motor torques in a joint space and compensating forfriction. The method may include distributing the motor torques in thejoint space to each motor before the secondary controller receives themotor torques for each motor.

In another aspect of the present disclosure, a controller for an endeffector that is controlled by four cables of an open loop differentialdrive mechanism includes a primary controller and a secondarycontroller. The primary controller is configured to receive a desiredpose for the end effector in yaw, pitch, and j aw degrees-of-freedom(DOF), to generate a motor torque for each motor of an instrument driveunit (IDU) to position the end effector in the desired pose, and totransmit the motor torques. The secondary controller is configured toreceive the motor torques from the primary controller, and to generatenull torques for each of the motors of the IDU to maintain tension inthe cables of the differential drive mechanism, and an IDU configured toreceive desired torques which include a sum of the motor torques and thenull torques, and to manipulate the end effector to the desired pose inresponse to receiving the desired torques.

In aspects, the controller includes a combination controller that isconfigured to receive the null torques and the motor torques from thesecondary controller, to generate the sum of the motor torques and nulltorques, and to transmit the sum to the IDU.

In another aspect of the present disclosure, an instrument drive unit(IDU) for controlling an end effector controlled by four cables of anopen loop differential drive mechanism includes motors, couplers, andtorque sensors. The IDU is controlled by a primary controller and asecondary controller. The motors are configured to receive desiredtorques and to manipulate the end effector to a desired pose in responseto receiving the desired torques. The primary controller is configuredto receive the desired pose for the end effector on yaw, pitch, and jawdegrees-of-freedom (DOF), to generate a motor torque for the motors toposition the end effector in the desired pose, and to transmit the motortorques. The secondary controller is configured to receive the motortorques from the primary controller, to generate null torques for themotors to maintain tension in the cables of the differential drivemechanism, wherein the desired torques include a sum of the motortorques and the null torques.

In another aspect of the present disclosure, an adapter for a surgicaltool that defines a longitudinal axis includes, a first drive screw, afirst drive nut, a first cable, a first spring, a second drive screw, asecond drive nut, a second cable, and a second spring. The first drivescrew is longitudinally fixed and configured to rotate about a firstscrew axis that is parallel to the longitudinal axis and has a threadedportion. The first drive nut is disposed about the threaded portion ofthe first drive screw and is threadably coupled to the first drive screwsuch that the first drive nut longitudinally translates in response torotation of the first drive screw and the first drive screw rotates inresponse to longitudinal translation of the first drive nut. The firstcable has a proximal portion fixed to the first drive nut and a distalportion. The first spring is disposed about the first drive screw andconfigured to urge the first drive nut in a first longitudinal directionand has a first spring constant.

The second drive screw is longitudinally fixed and configured to rotateabout a second screw axis that is parallel to the first screw axis andhas a threaded portion. The second drive nut is disposed about thethreaded portion of the second drive screw and is threadably coupled tothe second drive screw such that the second drive nut longitudinallytranslates in response to rotation of the second drive screw and thesecond drive screw rotates in response to longitudinal translation ofthe second drive nut. The second cable has a proximal portion fixed tothe second drive nut and a distal portion.

The distal portions of the first and second cables are operativelycoupled to one another such that translations of the distal portionsoppose one another. The second spring is disposed about the second drivescrew and configured to urge the second drive nut in the first directionand has a second spring constant. The second spring biasedsuch that thesecond spring translates the second drive nut and the second cable inthe first direction such that the second cable translates the firstcable and the first drive nut in a second direction opposite the firstdirection and against the bias of the first spring such that the tool isbiased towards a predetermined pose.

In aspects, the first screw includes a first proximal head that isconfigured to interface with a first motor and the second screw includesa second proximal head that is configured to interface with a secondmotor. The first direction may be proximal and the second direction maybe distal. The first drive nut may define a first slot with the proximalportion of the first cable fixed in the first slot. The second springconstant may be larger than the first spring constant.

In another aspect of the present disclosure a surgical tool includes anelongate shaft, an end effector, and an adapter. The elongated shaftdefines a longitudinal axis and has a proximal end and a distal end. Theend effector is supported adjacent the distal end of the elongate shaftand includes a first jaw and a second jaw movable in pitch, yaw, and jawDOFs. The adapter supports the proximal end of the elongate shaft andincludes a first drive screw, a first drive nut, a first cable, a firstspring, a second drive screw, a second drive nut, a second cable, and asecond spring. The first drive screw is longitudinally fixed andconfigured to rotate about a first screw axis that is parallel to thelongitudinal axis and has a threaded portion. The first drive nut isdisposed about the threaded portion of the first drive screw and isthreadably coupled to the first drive screw such that the first drivenut longitudinally translates in response to rotation of the first drivescrew and the first drive screw rotates in response to longitudinaltranslation of the first drive nut. The first cable extends through theelongate shaft and has a proximal portion fixed to the first drive nutand a distal portion secured to the end effector. The first spring isdisposed about the first drive screw and configured to urge the firstdrive nut in a first longitudinal direction and has a first springconstant.

The second drive screw is longitudinally fixed and configured to rotateabout a second screw axis that is parallel to the first screw axis andhas a threaded portion. The second drive nut is disposed about thethreaded portion of the second drive screw and is threadably coupled tothe second drive screw such that the second drive nut longitudinallytranslates in response to rotation of the second drive screw and thesecond drive screw rotates in response to longitudinal translation ofthe second drive nut. The second cable extends through the elongateshaft and has a proximal portion fixed to the second drive nut and adistal portion secured to the end effector.

The distal portions of the first and second cables are operativelycoupled to one another such that translations of the distal portionsoppose one another. The second spring is disposed about the second drivescrew and configured to urge the second drive nut in the first directionand has a second spring biased such that the second spring translatesthe second drive nut and the second cable in the first direction suchthat the second cable translates the first cable and the first drive nutin a second direction opposite the first direction and against the biasof the first spring such that the tool is biased towards a predeterminedpose.

In aspects, the distal portions of the first and second cables are eachcoupled to the first jaw.

In some aspects, the adapter includes a third drive screw, a third drivenut, a third cable, a third spring, a fourth drive screw, a fourth drivenut, a fourth cable, and a fourth spring. The third drive screw islongitudinally fixed within the adapter and configured to rotate about athird screw axis parallel to the longitudinal axis and has a threadedportion. The third drive nut is disposed about the threaded portion ofthe third drive screw and is threadably coupled to the third drive screwsuch that the third drive nut longitudinal translates in response torotation of the third drive screw and the third drive screw rotates inresponse to longitudinal translation of the third drive nut. The thirdcable extends through the elongate shaft and has a proximal portion thatis fixed to the third drive nut and a distal portion secured to the endeffector. The third spring is disposed about the third drive screw andis configured to urge the third drive nut in a third longitudinaldirection and has a third spring constant.

The fourth drive screw is longitudinally fixed within the adapter and isconfigured to rotate about a fourth screw axis that is parallel to thethird screw axis and has a threaded portion. The fourth drive nut isdisposed about the threaded portion of the fourth drive screw and isthreadably coupled to the fourth drive screw such that the fourth drivenut longitudinally translates in response to rotation of the fourthdrive screw and the fourth drive screw rotates in response tolongitudinal translation of the fourth drive nut. The fourth cableextends through the elongate shaft and has a proximal portion fixed tothe fourth drive nut and a distal portion secured to the end effector.The distal portions of the third and fourth cables are operativelycoupled to one another such that translations of the distal portionsoppose one another. The fourth spring is disposed about the fourth drivescrew, is configured to urge the fourth drive nut in the firstdirection, and has a fourth spring biased such that the fourth springtranslates the fourth drive nut and the fourth cable in the firstdirection. The fourth cable translating the third cable and the thirddrive nut in the second direction and against the bias of the thirdspring such that the end effector is biased towards the predeterminedpose.

In certain aspects, the distal portion of the first cable is secured toa first side of the first jaw, the distal portion of the second cable issecured to a second side of the first jaw, the distal portion of thethird cable is secured to the second side of the second jaw, and thedistal portion of the fourth cable is secured to the first side of thesecond jaw such that the first and fourth cables are disposed on thesame side of the first and second jaws, respectively, and the second andthird cables are disposed on the same side of the first and second jaws,respectively. The end effector may include a yoke and a clevis. Theclevis may be fixed to the distal end of the elongate shaft and the yokemay be pivotally coupled to the clevis about a first axis perpendicularto and intersected by the longitudinal axis. The jaw may be pivotallycoupled to the yoke about a second axis perpendicular to the first axis.The first jaw may have a first spindle pivotal about the second axis andthe second jaw may have a second spindle pivotal about the second axis.The distal portions of the first and second cables may be secured toopposite sides of the first spindle and the distal portions of the thirdand fourth cables may be secured to opposite sides of the secondspindle.

In particular aspects, the second and fourth springs are configured tomaintain the tool in a pose with the first and second jaws in a closedposition, the first and second jaws longitudinally aligned with thelongitudinal axis, and the yoke aligned with the longitudinal axis. Thefirst, second, third, and fourth cables may be configured to manipulatethe pose of the end effector in pitch, yaw, and jaw DOFs.

In another aspect of the present disclosure, a surgical tool configuredto selectively connect to a drive unit includes an elongate shaft, anend effector, a first cable, a second cable, a third cable, a fourthcable, and an adapter. The elongate shaft defines a longitudinal axisand has a proximal end and a distal end. The end effector is supportedadjacent the distal end of the elongate shaft and includes a first jawand a second jaw. The first cable extends through the elongate shaft andhas a distal portion secured to a first side of the first jaw. Thesecond cable extends through the elongate shaft and has a distal portionsecured to a second, opposite side of the first jaw. The third cableextends through the elongate shaft and has a distal portion secured tothe second side of the second jaw.

The fourth cable extends through the elongate shaft and has a distalportion secured to the first side of the second jaw. The adaptersupports the proximal end of the elongate shaft and is configured toselectively connect to a drive unit. The adapter includes a differentialdrive mechanism configured to manipulate proximal portions of each ofthe first, second, third, and fourth cables to manipulate the endeffector in pitch, yaw, and jaw DOFs. Each of the first, second, third,and fourth cables are biased proximally and configured to maintain theend effector in a desired pose with the tool is disconnected from adrive unit.

In aspects, the adapter urges each of the first and third cablesproximally with a first force and urges each of the second and fourthcable proximally with a second force greater than the first force. Thedesired pose may be straight in pitch and yaw with the first and secondjaws in a closed position such that the first and second jaws are closedand aligned with the longitudinal axis. Alternatively, the desired posemay be straight in pitch and yaw with the first and second jaws in anopen position such that the first and second jaws are spaced apart fromone another and aligned with the longitudinal axis.

In some aspects, the end effector includes a yoke and a clevis. Theclevis may be fixed to the distal end of the elongate shaft. The yoke ispivotally coupled to the clevis about a first axis that is perpendicularto and intersected by the longitudinal axis. The jaws may be pivotallycoupled to the yoke about a second axis perpendicular to the first axis.The first jaw has a first spindle pivotal about the second axis and thesecond jaw has a second spindle pivotal about the second axis.

In certain aspects, the differential drive mechanism includes a drivescrew, a nut threadably coupled to the drive screw, and a spring biasingthe nut proximally for each of the first, second, third, and fourthcables with the proximal portion of each of the first, second, third,and fourth cables fixed to a respective nut.

Further, to the extent consistent, any of the aspects described hereinmay be used in conjunction with any or all of the other aspectsdescribed herein.

BRIEF DESCRIPTION OF THE DRAWINGS

Various aspects of the present disclosure are described hereinbelow withreference to the drawings, which are incorporated in and constitute apart of this specification, wherein:

FIG. 1 is a schematic illustration of a user interface and a surgicalrobot of a robotic surgical system in accordance with the presentdisclosure;

FIG. 2 is a side, perspective view of an arm of the robotic system ofFIG. 1 including an IDU, an adapter assembly, and a tool having an endeffector;

FIG. 3 is a rear, perspective view of the tool of FIG. 2;

FIG. 4 is a cross-sectional view taken along the section line 3-3 ofFIG. 3;

FIG. 5 is a perspective view showing internals of an adapter of the toolof FIG. 3;

FIG. 6 is an enlarged, perspective view of the end effector of the toolof FIG. 2 with jaws in a straight, closed configuration;

FIG. 7 is a perspective view of the end effector of FIG. 6 with the endeffector rotated in a positive yaw DOF;

FIG. 8 is a perspective view of the end effector of FIG. 6 with the endeffector rotated in a positive pitch DOF;

FIG. 9 is a perspective view of the end effector of FIG. 6 with the endeffector rotated in a positive jaw DOF;

FIG. 10 is a perspective view of the end effector of FIG. 6 with the endeffector rotated in the positive pitch DOF, the positive yaw DOF, andthe positive jaw DOF;

FIG. 11 is a schematic of a positional controller in accordance with anembodiment of the present disclosure;

FIG. 12 is a schematic of an overall controller in accordance with anembodiment of the present disclosure;

FIG. 13 is a schematic of another overall controller in accordance withan embodiment of the present disclosure; and

FIG. 14 is a schematic of another overall controller in accordance withan embodiment of the present disclosure.

DETAILED DESCRIPTION

Embodiments of the present disclosure are now described in detail withreference to the drawings in which like reference numerals designateidentical or corresponding elements in each of the several views. Asused herein, the term “clinician” refers to a doctor, a nurse, or anyother care provider and may include support personnel. Throughout thisdescription, the term “proximal” refers to the portion of the device orcomponent thereof that is closer to the clinician or surgical robotmanipulating the device or component and the term “distal” refers to theportion of the device or component thereof that is farther from theclinician or surgical robot manipulating the device.

Referring to FIG. 1, a robotic surgical system 1 in accordance with thepresent disclosure is shown generally as a robotic system 10, aprocessing unit 30, and a user interface 40. The robotic system 10generally includes linkages or arms 12 and a robot base 18. The arms 12moveably support a tool 20 which is configured to act on tissue. Thearms 12 each have an end 14 that supports a tool 20 which is configuredto act on tissue. In addition, the ends 14 of the arms 12 may include animaging device 16 for imaging a surgical site. The user interface 40 isin communication with robot base 18 through the processing unit 30.

The user interface 40 includes a display device 44 which is configuredto display three-dimensional images. The display device 44 displaysthree-dimensional images of the surgical site which may include datacaptured by imaging devices 16 positioned on the ends 14 of the arms 12and/or include data captured by imaging devices that are positionedabout the surgical theater (e.g., an imaging device positioned withinthe surgical site, an imaging device positioned adjacent the patient,imaging device 56 positioned at a distal end of an imaging linkage orarm 52). The imaging devices (e.g., imaging devices 16, 56) may capturevisual images, infra-red images, ultrasound images, X-ray images,thermal images, and/or any other known real-time images of the surgicalsite. The imaging devices transmit captured imaging data to theprocessing unit 30 which creates three-dimensional images of thesurgical site in real-time from the imaging data and transmits thethree-dimensional images to the display device 44 for display.

The user interface 40 also includes input handles 42 which are supportedon control arms 43 which allow a clinician to manipulate the roboticsystem 10 (e.g., move the arms 12, the ends 14 of the arms 12, and/orthe tools 20). Each of the input handles 42 is in communication with theprocessing unit 30 to transmit control signals thereto and to receivefeedback signals therefrom. Additionally or alternatively, each of theinput handles 42 may include input devices (not shown) which allow thesurgeon to manipulate (e.g., clamp, grasp, fire, open, close, rotate,thrust, slice, etc.) the tools 20 supported at the ends 14 of the arms12.

Each of the input handles 42 is moveable through a predefined workspaceto move the ends 14 of the arms 12 within a surgical site. Thethree-dimensional images on the display device 44 are orientated suchthat movement of the input handle 42 moves the ends 14 of the arms 12 asviewed on the display device 44. It will be appreciated that theorientation of the three-dimensional images on the display device may bemirrored or rotated relative to a view from above the patient. Inaddition, it will be appreciated that the size of the three-dimensionalimages on the display device 44 may be scaled to be larger or smallerthan the actual structures of the surgical site permitting a clinicianto have a better view of structures within the surgical site. As theinput handles 42 are moved, the tools 20 are moved within the surgicalsite as detailed below. As detailed herein, movement of the tools 20 mayalso include movement of the ends 14 of the arms 12 which support thetools 20.

For a detailed discussion of the construction and operation of a roboticsurgical system 1, reference may be made to U.S. Pat. No. 8,828,023, theentire contents of which are incorporated herein by reference.

With reference to FIG. 2, a portion of an exemplary arm 12 of thesurgical robot 10 of FIG. 1. The arm 12 includes a carriage 122 that istranslatable along a rail 124. An instrument drive unit (IDU) 13 issecured to the carriage 122. The IDU 13 has one or more motors (notshown) that are configured to control a tool 20 as detailed below. For adetailed discussion of an exemplary IDU including one or more motors,reference may be made to U.S. Patent Publication No. 2018/0153634, theentire contents of which are incorporated herein by reference.

The tool 20 includes an adapter 210, an elongate shaft 212 that extendsdistally from the adapter 210, and an end effector 270 supported by adistal portion of the elongate shaft 212. The adapter 210 is releasablycoupled to the IDU 13 such that the tool 20 receives input from the IDU13.

With additional reference to FIG. 3, the adapter 210 includes an IDUinterface 220 including a first motor interface 222, a second motorinterface 224, a third motor interface 226, a fourth motor interface228, and an control interface 229. Each of the motor interfaces 222-228is configured to mechanically couple to a respective motor of the IDU13. The motor interfaces 222, 224, 226, 228 are arranged about thelongitudinal axis A-A of the shaft 212. The motor interfaces 222, 224,226, 228 may from a rectangle or square in a plane orthogonal to thelongitudinal axis A-A of the shaft 212. The control interface 229 isconfigured to couple to a control interface of the IDU 13 or thecarriage 122 to receive instructions from the surgical robot 10 and/orthe processing unit 30 and/or to transmit data to the surgical robot 10and/or the processing unit 30.

Referring to FIGS. 4 and 5, the first motor interface 222 includes afirst drive screw 230, a first drive nut 232, a first spring 234, and afirst cable 236. The first drive screw 230 includes a first head 231 anda distal nub 239. The first head 231 may have radial teeth, a slot, afemale connector, a male connector, or any suitable interface forcoupling coaxial rotating shafts such that the first head 231 isconfigured to mechanically couple the first drive screw 230 to a motorof the IDU 13. The first drive screw 230 is supported within the adapter210 by a first bearing 233 positioned adjacent the first head 231. Thedistal nub 239 is received within a first opening 238 defined by theadapter 210 such that the first bearing 233 and the distal nub 239support the first drive screw 230 within the adapter 210 and enable thefirst drive screw 230 to rotate about its longitudinal axis, maintainthe longitudinal axis of the first drive screw 230 parallel to alongitudinal axis of the elongate shaft 212, and prevent the first drivescrew 230 from translating along its longitudinal axis.

The first drive nut 232 is disposed over a threaded portion of the firstdrive screw 230 such that the first drive nut 232 and the first drivescrew 230 are threadably coupled with one another. Specifically, as thefirst drive screw 230 is rotated in a first direction, e.g., clockwiserotation about the longitudinal axis of the drive screw as shown witharrow D₁ in FIG. 3, the first drive nut 232 translates proximally alongthe first drive screw 230 towards the first head 231 and when the firstdrive screw 230 is rotated in a second direction opposite the firstdirection, e.g., counter-clockwise, the first drive nut 232 istranslated distally along the first drive screw 230 away from the firsthead 231. The first drive nut 232 defines a first slot 235 that receivesa portion of the first cable 236 such that as the first drive nut 232translates along the first drive screw 230, the first cable 236cooperates with translation of the first drive nut 232. Specifically,the first cable 236 is retracted as the first drive nut 232 translatesproximally and the first cable 236 is relaxed as the first drive nut 232translates distally.

The first spring 234 is disposed about the first drive screw 230 distalof the first drive nut 232 and engages a distal surface of the firstdrive nut 232. The first spring 234 is supported within the adapter 210such that the first spring 234 urges the first drive nut 232 proximally.The first spring 234 may have a spring constant large enough to urge thefirst drive nut 232 proximally such that the first drive screw 230 isrotated in the first direction absent a force applied to the first head231. The first spring 234 may have a constant or progressive springconstant.

The second motor interface 224 includes a second drive screw 240, asecond spring 244, and a second cable 246; the third motor interface 226includes a third drive screw 250, a third spring 254, and a third cable256; and the fourth motor interface 228 includes a fourth drive screw260, a fourth spring 264, and a fourth cable 266. The drive screws 240,250, 260, the springs 244, 254, 264, and the cables 246, 256, 266 aresimilar to the first drive screw 230, the first spring 234, and thefirst cable 236, respectively, detailed above and will not be detailedherein for brevity except where the differences are relevant to thefunction of the tool 20.

With reference to FIGS. 6-10, the cables 236, 246, 256, 266 extendthrough the shaft 212 and are connected to the end effector 270 tocontrol movement of the end effector 270 in three degrees-of-freedom(DOF), e.g., yaw, pitch, and jaw. The end effector 270 includes a clevis272, a yoke 274, a first jaw 276, and a second j aw 278. The clevis 272includes a first idler 273 and the yoke 274 includes a second idler 275distal of the first idler 273. The first and second idlers 273, 275 eachdefine an idler axis I₁, I₂ that is perpendicular to the longitudinalaxis A-A of the shaft 212 and parallel to one another.

The first jaw 276 includes a first spindle 277 and the second jaw 278includes a second spindle 279. The first spindle 277 and the secondspindle 279 each define a spindle axis S₁, S₂ that is perpendicular tothe longitudinal axis A-A of the shaft 212, when the shaft 212 is in astraight configuration as shown in FIG. 5, and perpendicular to thesecond idler axis I₂. The first and second spindle axes S₁, S₂ may becoaxial with one another.

The clevis 272 pivotally supports the yoke 274 about the second idleraxis I₂ in a yaw DOF. The yoke 274 pivotally supports the first andsecond jaws 276, 278 about the first and second spindle axes S₁, S₂ inpitch and jaw. Specifically, when the first and second jaws 276, 278pivot about the first and second spindle axes S₁, S₂ in the samedirection in concert with one another, the first and second jaws 276,278 move in a pitch DOF. Alternatively, when the first and second jaws276, 278 pivot about the first and second spindle axes S₁, S₂ inopposite directions or independent of one another, the first and secondjaws 276, 278 move in a jaw DOF. The first and second jaws 276, 278 canmove in the same direction but at different speeds, e.g., not inconcert, such that the first and second jaws 276, 278 move in both thepitch DOF and the jaw DOF.

Continuing to refer to FIG. 6, the cables 236, 246, 256, 266 wrap aroundthe first and second idlers 273, 275 and are secured to a respective oneof the first and second spindles 277, 279. The first and second cables236, 246 are secured to opposite sides of the first spindle 277 of thefirst jaw 276. Specifically, the first cable 236 may be secured to a topside of the first spindle 277 and the second cable 246 may be secured toa bottom side of the first spindle 277. The first and second cables 236,246 may form a continuous monolithic cable with one another that wrapsabout the first spindle 277. The third and fourth cables 256, 266 aresecured to opposite sides of the second spindle 279 of the second jaw278. Specifically, the third cable 256 may be secured to a bottom sideof the second spindle 279 and the fourth cable 266 may be secured to abottom side of the second spindle 279. The third and fourth cables 256,266 may form a continuous monolithic cable with one another that wrapsabout the second spindle 279.

The first and second idlers 273, 275 may define a separate groove toguide each of the cables 236, 246, 256, 266 around the respective idler273, 275 such that the cables 236, 246, 256, 266 are secured within therespective groove without interfering with the other cables 236, 246,256, 266.

Continuing to refer to FIGS. 6-10, displacement of the cables 236, 246,256, 266 are used as a differential drive to control the end effector270 in the yaw DOF, the pitch DOF, and the jaw DOF. Initially referringto FIG. 6, the end effector 270 is in a straight configuration in whichthe cables 236, 246, 256, 266 are in a neutral position relative to oneanother. In the neutral position, the tension in each of the cables 236,246, 256, 266 may be substantially equal to one another. To move the endeffector 270 in a positive yaw direction, the third and fourth cables256, 266, which are each secured to the second spindle 278, are eachretracted a distance and the first and second cables 236, 246, which areeach secured to the first spindle 276, are each extended or relaxed thesame distance as shown in FIG. 7. As shown, the positive yaw directionis pivoting the yoke 274 to the right and the negative yaw direction ispivoting the yoke 274 to the left. To move the end effector 270 in thenegative yaw direction, the first and second cables 236, 246 areretracted and the third and fourth cables 256, 266 are extended orrelaxed the same distance that the first and second cables 236, 246 areretracted.

Referring now to FIG. 8, to move the end effector 270 in a positivepitch direction, upwardly as shown, the first and fourth cables 236,266, which are each secured to the top side of different spindles 276,278, are retracted a distance and the second and third cables 246, 256,which are each secured to the bottom side of different spindles 276,278, are relaxed the same distance. To move the end effector 270 in thenegative pitch direction, downwardly as shown, the second and thirdcables 246, 256 are retracted a distance and the first and fourth cables236, 266 are relaxed the same distance.

With reference to FIG. 9, to move the end effector 270 in a positive jawdirection, to pivot the jaws 276, 278 apart from one another, the firstand third cables 236, 256, which are each secured to different sides ofdifferent spindles 276, 278, are retracted and the second and fourthcables 246, 266, which are secured to different sides of differentspindles 276, 278, are relaxed. The first and third cables 236, 256 maybe retracted the same distances or different distances. However, thesecond cable 246 is relaxed the same distance that the first cable 236is retracted and the fourth cable 266 is relaxed the same distance thatthe third cable 256 is retracted. To move the end effector 270 in thenegative jaw direction, e.g., to move the jaws 276, 278 towards oneanother, the second and fourth cables 246, 266 are retracted and thefirst and third cables 236, 256 are relaxed.

It is contemplated that one jaw, e.g., second jaw 278, may be stationaryas the other jaw, e.g., first jaw 276, is pivoted such that the endeffector 270 moves in the jaw direction. To move the end effector 270 inthe positive jaw direction with second jaw 278 stationary, the firstcable 236 can be retracted and the second cable 246 is relaxed an equalamount while the third and fourth cables 256, 266 remain stationary.Similarly to move the end effector 270 in the positive jaw directionwith the first jaw 276 stationary, the third cable 236 is retracted andthe fourth cable 266 is relaxed and equal amount while the first andsecond cables 236, 246 remain stationary. Each of these movements can bereversed to move the end effector 270 in the negative jaw direction withone of the jaws 276, 278 stationary.

Referring now to FIG. 10, the end effector 270 may be moved in more thanone DOF sequentially or simultaneously. For example, the end effector270 may be moved from the straight configuration (FIG. 6) to theposition in FIG. 10 by retracting the fourth cable 266 and relaxing thesecond cable 246 while substantially maintaining the position of thefirst and third cables 236, 256 to move the end effector 270 in thepositive yaw, pitch, and jaw directions simultaneously.

While several movements of the end effector 270 in the yaw DOF, thepitch DOF, and the jaw DOF are described above, these are meant to beexemplary movements and not an exhaustive list of all possible movementsor combination of movements of the end effector 270 in the yaw, pitch,and jaw DOFs.

Referring to FIGS. 2, 5, and 6, it may be desirable to maintain the endeffector 270 in a known or neutral position when the tool 20 isdisconnected from the IDU 13. By maintaining the tool 20 in a knownposition, the robotic system 1 can know the position or pose of the endeffector 270 when the tool 20 is connected to the IDU 13 without theneed for running a calibration sequence. This may reduce the timerequired to calibrate a surgical robot 10 each time a new tool 20 isattached. In addition, the robotic system 10 can know the position ofthe end effector 270 without requiring absolute encoders which mayreduce the cost of each tool 20. This known pose may be stored in memory(not shown) of the tool 20 and communicated to the robotic system 1through the control interface 29 when the tool 20 is attached to thesurgical robot 10.

To maintain the end effector 270 in a known or neutral pose, the springs234, 244, 254, 264 can be used as pretension springs. For some tools,e.g., clip appliers or staplers, it may be beneficial to have endeffectors maintained in a fully open, or fully positive jaw,configuration when the tool 20 is disconnected from the IDU 13. Forexample, clip appliers and staplers may need to be in an open positionto load clips or staples into a jaw of the end effector. For suchinstruments, the first and third springs 234, 254 each have a large,first spring constant and the second and fourth springs 244, 264 eachhave a smaller, second spring constant such that the first and thirdsprings 234, 254 overpower the second and fourth springs 244, 264 tomove the jaws 276, 278 towards the fully open configuration. Inaddition, as the second and fourth springs 244, 264 maintain tension inthe second and third cables 246, 266, such that the end effector 270remains straight with respect to the yaw and pitch directions.Additionally or alternatively, the first and third springs 234, 254 mayhave the same or substantially the same spring constant as the secondand fourth springs 244, 264 and be biased such that the first and thirdsprings 234, 254 overpower the second and fourth springs 244, 264 tomove the jaws towards the fully open configuration.

Alternatively, it may be beneficial for some end effectors to bemaintained in a fully closed, or fully negative jaw, configuration whenthe tool 20 is disconnected from the IDU 13. For such instruments, thesecond and fourth springs 244, 264 each have a large, first springconstant and the first and third springs 234, 254 each have a smaller,second spring constant such that the second and fourth springs 244, 264overpower the first and third springs 234, 254 to move the jaws 276, 278towards the full closed configuration. In addition, as the first andthird springs 234, 254 maintain tension in the first and third cables236, 256, the end effector 270 remains straight with respect to the yawand pitch directions. Additionally or alternatively, the second andfourth springs 244, 264 may have the same or substantially the samespring constant as the first and third springs 234, 254 and be biasedsuch that the second and fourth springs 244, 264 overpower the first andthird springs 234, 254 to move the jaws towards the fully closedconfiguration.

The springs 234, 244, 254, 256 may be configured to maintain the tool 20in other neutral positions which may be beneficial for a tool 20 with aparticular end effector 270. The configuration of the tool 20 may bemaintained by varying spring constants of one or more of the springs234, 244, 254, 256 and/or biasing one or more of the springs 234, 244,254, 256. The neutral position of the tool 20 may be communicated to thesurgical robot 10 and/or the processing unit 30 through the controlinterface 229, when the tool 20 is connected to the IDU 13.

A forward kinematic model to relate measured motor positions of themotors of the IDU 13 to calculated yaw, pitch, and jaw positions of theend effector 270 is required to determine a pose of the end effector270. This is different than traditional robots where joint angles aregenerally directly measured by position sensors, such as potentiometersor encoders. However, as space is extremely limited within the endeffector 270, it is difficult to place encoders, potentiometers, orother devices which directly measure the pose of the end effector 270within the end effector 270. Thus, it is advantageous to calculate thepose of the end effector from the measured position of the motors of theIDU 13. In addition, inaccuracies of the calculated pose can becompensated for by observations of the end effector 270 within thesurgical site by a clinician interfacing with the robotic surgicalsystem 1 (FIG. 1).

Kinematic control methods for controlling the end effector 270 in theyaw DOF, the pitch DOF, and the jaw DOF are described below withreference to the tool 20 detailed in FIGS. 2-6. In the model below, thefirst jaw 276 will be referred to as jaw a and the second jaw 278 willbe referred to as jaw b to avoid confusion with integers used in thefollowing equations. It will be appreciated that the arm 12 of thesurgical robot 10 is configured to move the end effector 270 in anadditional four DOFs such that the end effector 270 is moveable in sixDOFs and the jaw DOF.

As detailed above, the jaws “a, b” are moved by displacing one or moreof the cables 236, 246, 256, 266 with the motors (not shown) of the IDU13. A first torque τ_(a) is the torque applied to the first jaw “a” anda second torque τ_(b) is the torque applied to the second jaw “b”. Asshown, the first and second jaws “a, b” are typically mirror images ofeach other with only minor differences and both of the first and secondjaws “a, b” pivot about the same pin or axis, e.g., spindle axes S₁, S₂.As such, the first and second torques τ_(a), τ_(b) can be represented bythe following equations:

$\begin{matrix}\left\{ \begin{matrix}{\tau_{a} = {{m_{a}{\overset{¨}{\theta}}_{a}} + {c_{a}{\overset{.}{\theta}}_{a}} + {{\mu_{1}\left( {f_{1} + f_{2}} \right)}{{sgn}\left( {\overset{.}{\theta}}_{a} \right)}r_{p}}}} \\{\tau_{b} = {{m_{b}{\overset{¨}{\theta}}_{b}} + {c_{b}{\overset{.}{\theta}}_{b}} + {{\mu_{1}\left( {f_{3} + f_{4}} \right)}{{sgn}\left( {\overset{.}{\theta}}_{b} \right)}r_{p}}}}\end{matrix} \right. & (1)\end{matrix}$

where μ₁ is the coefficient of friction between the respective jaw “a,b” and the pin (not explicitly shown) that the respective jaw “a, b” ispivotal about, c_(a), c_(b) are dampening terms, and “m” is an inertialterm. The dampening terms c_(a), c_(b) and the inertial term “m” are afunction of a joint angle θ.

In addition, the first jaw a is directly articulated by the first andsecond cables 236, 246 and the second jaw b is directly articulated bythe third and fourth cables 256, 266 with the first and fourth cables236, 266 being secured to the top side of the respective jaw “a, b”, andthe second and third cables 246, 256 being secured to the bottom side ofthe respective jaw “a, b”. As such the torque τ_(a), τ_(b) can also berepresented by the following equations:

$\begin{matrix}\left\{ \begin{matrix}{\tau_{a} = {\left( {f_{1} - f_{2}} \right)r_{p}}} \\{\tau_{b} = {\left( {f_{4} - f_{3}} \right)r_{p}}}\end{matrix} \right. & (2)\end{matrix}$

If each jaw “a, b” is considered separately, the inertial term “m” andthe dampening term “c” are independent of the joint angle θ. Since thejaws “a, b”, are substantial mirrors of each other, it can be assumedthat the combined term of the inertial term “m” and the dampening term“c” is twice the respective term from a single jaw such thatm_(p)≡2m_(a)≈2m_(b) and c_(p)≡2c_(a)≈2c_(b). Thus, combining Equation(2) into Equation (1), and rearranging the equation, results in thefollowing:

$\begin{matrix}\left\{ \begin{matrix}\begin{matrix}{{\left( {f_{1} - f_{2}} \right)r_{p}} - {{\mu_{1}\left( {f_{1} + f_{2}} \right)}{{sgn}\left( {\overset{.}{\theta}}_{a} \right)}r_{p}} + {\left( {f_{4} - f_{3}} \right)r_{p}} -} \\{{{\mu_{1}\left( {f_{3} + f_{4}} \right)}{{sgn}\left( {\overset{.}{\theta}}_{b} \right)}r_{p}} = {{m_{p}\frac{{\overset{¨}{\theta}}_{a} + {\overset{¨}{\theta}}_{b}}{2}} + {c_{p}\frac{{\overset{.}{\theta}}_{a} + {\overset{.}{\theta}}_{b}}{2}}}}\end{matrix} \\{{2\begin{Bmatrix}{{\left( {f_{1} - f_{2}} \right)r_{p}} - {{\mu_{1}\left( {f_{1} + f_{2}} \right)}{{sgn}\left( {\overset{.}{\theta}}_{a} \right)}r_{p}} -} \\{{\left( {f_{4} - f_{3}} \right)r_{p}} + {{\mu_{1}\left( {f_{3} + f_{4}} \right)}{{sgn}\left( {\overset{.}{\theta}}_{b} \right)}r_{p}}}\end{Bmatrix}} = {{m_{p}\left( {{\overset{¨}{\theta}}_{a} - {\overset{¨}{\theta}}_{b}} \right)} + {c_{p}\left( {{\overset{.}{\theta}}_{a} - {\overset{.}{\theta}}_{b}} \right)}}}\end{matrix} \right. & (3)\end{matrix}$

Next, a synthetic joint of pitch, for the pitch DOF, is formed bycombining jaw “a” and jaw “b”. As noted above, the inertial term “m_(p)”and the dampening term “c_(p)” of the synthetic pitch joint are twicethat of the dampening terms and inertial terms for each of theindividual jaws “a, b”. The pitch angle θ_(p) is defined by a linebisecting an angle between the first and second jaws “a, b”. thus, thearticulation torque in pitch is defined as:

$\begin{matrix}\left\{ \begin{matrix}\begin{matrix}{\tau_{p} \equiv {{\left( {f_{1} - f_{2}} \right)r_{p}} - {{\mu_{1}\left( {f_{1} + f_{2}} \right)}{{sgn}\left( {\overset{.}{\theta}}_{a} \right)}r_{p}} +}} \\{{\left( {f_{4} - f_{3}} \right)r_{p}} - {{\mu_{1}\left( {f_{3} + f_{4}} \right)}{{sgn}\left( {\overset{.}{\theta}}_{b} \right)}r_{p}}}\end{matrix} \\{\theta_{p} \equiv \frac{\theta_{a} + \theta_{b}}{2}}\end{matrix} \right. & (4)\end{matrix}$

A synthetic joint of jaw, for the jaw DOF, is formed by the combinationof the first and second jaws “a, b” with joint pitch as:

$\begin{matrix}\left\{ \begin{matrix}{\tau_{j}\  \equiv {2\begin{Bmatrix}{{\left( {f_{1} - f_{2}} \right)r_{p}} - {{\mu_{1}\left( {f_{1} + f_{2}} \right)}{{sgn}\left( {\overset{.}{\theta}}_{a} \right)}r_{p}} -} \\{{\left( {f_{4} - f_{3}} \right)r_{p}} + {{\mu_{1}\left( {f_{3} + f_{4}} \right)}{{sgn}\left( {\overset{.}{\theta}}_{b} \right)}r_{p}}}\end{Bmatrix}}} \\{\theta_{j} \equiv {\theta_{a} - \theta_{b}}}\end{matrix} \right. & (5)\end{matrix}$

Substituting Equations (3) and (4) into Equation (2) provides basicdynamic equations for the synthetic joint of pitch and the syntheticjoint of jaw as:

$\begin{matrix}\left\{ \begin{matrix}{\tau_{p} = {{m_{p}{\overset{¨}{\theta}}_{p}} + {c_{p}{\overset{.}{\theta}}_{p}}}} \\{\tau_{j} = {{m_{p}{\overset{¨}{\theta}}_{j}} + {c_{p}{\overset{.}{\theta}}_{j}}}}\end{matrix} \right. & (6)\end{matrix}$

As detailed above, Equations (4) and (5) describe the relationshipbetween forces in the cables 236, 246, 256, 266 and the articulationtorque for the synthetic joints of pitch and jaw. The final DOF of theend effector 270 is the yaw DOF which is articulated by all four cables.For the joint of yaw, the dynamic equation depends on the configurationof the end effector 270 in pitch and jaw. Generally, in medicalapplications the motion in pitch and jaw are quick. As such, forsimplicity, this dependency can be ignored to provide:

(f ₁ +f ₂)r _(y)−(f ₄ +f ₃)r _(y) =m _(y){umlaut over (θ)}_(y) +c_(y){dot over (θ)}_(y)+μ₂(f ₁ +f ₂ +f ₃ +f ₄)sgn({dot over (θ)}_(y))r_(y)  (7)

where inertial term “m” and Coriolis and centrifugal term “c” are lumpedparameters that take the pitch and jaw joints into account and μ₂ is afriction coefficient of the yoke 274 about the idler 275. Equation (7)is rearranged to define articulating torque of yaw τ_(y) as:

τ_(y)≡(f ₁ +f ₂)r _(y)−(f ₄ +f ₃)r _(y)−μ₂(f ₁ +f ₂ +f ₃ +f ₄)sgn({dotover (θ)}_(y))r _(y)  (8)

Equation (8) can be substituted into Equation (7) to simplify a dynamicequation for the yaw joint as:

τ_(y) =m _(y){umlaut over (θ)}_(y) +c _(y){dot over (θ)}_(y)  (9)

Thus the dynamic equations for the yaw, pitch, and jaw joints are:

$\begin{matrix}\left\{ \begin{matrix}{\tau_{y} = {{m_{y}{\overset{¨}{\theta}}_{y}} + {c_{y}{\overset{.}{\theta}}_{y}}}} \\{\tau_{p} = {{m_{p}{\overset{¨}{\theta}}_{p}} + {c_{p}{\overset{.}{\theta}}_{p}}}} \\{\tau_{j} = {{m_{p}{\overset{¨}{\theta}}_{j}} + {c_{p}{\overset{.}{\theta}}_{j}}}}\end{matrix} \right. & (10)\end{matrix}$

Further, a set of equations to relate articulating torques to forces inthe cables 236, 246, 256, 266 can be determined from Equations (4), (5),and (8) such that:

$\quad\begin{matrix}\left\{ \begin{matrix}{\tau_{y} \equiv {{\left( {f_{1} + f_{2}} \right)r_{y}} - {\left( {f_{4} + f_{3}} \right)r_{y}} - {{\mu_{2}\left( {f_{1} + f_{2} + f_{3} + f_{4}} \right)}{{sgn}\left( {\overset{.}{\theta}}_{y} \right)}r_{y}}}} \\\begin{matrix}{\tau_{p} \equiv {{\left( {f_{1} - f_{2}} \right)r_{p}} - {{\mu_{1}\left( {f_{1} + f_{2}} \right)}{{sgn}\left( {\overset{.}{\theta}}_{a} \right)}r_{p}} +}} \\{{\left( {f_{4} - f_{3}} \right)r_{p}} - {{\mu_{1}\left( {f_{3} + f_{4}} \right)}{{sgn}\left( {\overset{.}{\theta}}_{b} \right)}r_{p}}}\end{matrix} \\{\tau_{j} \equiv {2\begin{Bmatrix}{{\left( {f_{1} - f_{2}} \right)r_{p}} - {{\mu_{1}\left( {f_{1} + f_{2}} \right)}{{sgn}\left( {\overset{.}{\theta}}_{a} \right)}r_{p}} -} \\{{\left( {f_{4} - f_{3}} \right)r_{p}} + {{\mu_{1}\left( {f_{3} + f_{4}} \right)}{{sgn}\left( {\overset{.}{\theta}}_{b} \right)}r_{p}}}\end{Bmatrix}}}\end{matrix} \right. & (11)\end{matrix}$

This can be simplified by defining the following variables:

$\begin{matrix}\left\{ \begin{matrix}{\eta_{1a} \equiv {\mu_{1}{{sgn}\left( {\overset{.}{\theta}}_{a} \right)}}} \\{\eta_{1b} \equiv {\mu_{1}{{sgn}\left( {\overset{.}{\theta}}_{b} \right)}}} \\{\eta_{2} \equiv {\mu_{2}{{sgn}\left( {\overset{.}{\theta}}_{y} \right)}}}\end{matrix} \right. & (12)\end{matrix}$

Such that Equations (11) can be expressed in matrix form as:

$\begin{matrix}{\begin{bmatrix}\tau_{y} \\\tau_{p} \\\tau_{j}\end{bmatrix} = {\quad{\begin{bmatrix}{r_{y}\left( {1 - \eta_{2}} \right)} & {r_{y}\left( {1 - \eta_{2}} \right)} & {- {r_{y}\left( {1 + \eta_{2}} \right)}} & {- {r_{y}\left( {1 + \eta_{2}} \right)}} \\{r_{p}\left( {1 - \eta_{1a}} \right)} & {- {r_{p}\left( {1 + \eta_{1a}} \right)}} & {- {r_{p}\left( {1 + \eta_{1b}} \right)}} & {r_{p}\left( {1 - \eta_{1b}} \right)} \\{2{r_{p}\left( {1 - \eta_{1a}} \right)}} & {{- 2}{r_{p}\left( {1 + \eta_{1a}} \right)}} & {2{r_{p}\left( {1 + \eta_{1b}} \right)}} & {{- 2}{r_{p}\left( {1 - \eta_{1b}} \right)}}\end{bmatrix}\begin{bmatrix}f_{1} \\f_{2} \\f_{3} \\f_{4}\end{bmatrix}}}} & (13)\end{matrix}$

An inverse kinematic model can be formed to determine desired yaw,pitch, and jaw angles of the end effector 270 to motor positions foreach the motors of the IDU 13. As detailed above, movements in the yaw,pitch, and jaw DOFs of the end effector 270 can be achieved with thedifferential drive mechanism of the adapter 210. Further, movement inthe negative of each of the DOF uses the same cables, e.g., cables 236,246, 256, 266, but with the opposite direction or sign on each of thecables. It is noted that a negative sign on movement of a cable does notnecessarily represent retraction of the cable but that the respectivecable is kept under minimal tension to prevent the cable from goingslack. Table 1, below, shows the combinations of movement in thepositive direction in each of the DOFs. It is noted that translations ofthe jaw DOF are half due to each jaw “a, b” moving and contributing tothe total jaw angle.

TABLE 1 Movement Cable 236 Cable 246 Cable 256 Cable 266 Yaw −s_(y)−s_(y) +s_(y) +s_(y) Pitch +s_(p) −s_(p) −s_(p) +s_(p) Jaw +0.5s_(j)−0.5s_(j) +0.5s_(j) −0.5s_(j)

The movement of each of the cables in a linear combination of the cablemovements in yaw, pitch, and jaw such that:

$\begin{matrix}\left\{ \begin{matrix}{s_{1} = {{- s_{y}} + s_{p} + {0.5s_{j}}}} \\{s_{2} = {{- s_{y}} - s_{p} - {0.5s_{j}}}} \\{s_{3} = {s_{y} - s_{p} + {0.5s_{j}}}} \\{s_{4} = {s_{y} + s_{p} - {0.5s_{j}}}}\end{matrix} \right. & (14)\end{matrix}$

Thus, the movement of each cable 236, 246, 256, 266 for each of themovements can be given by:

$\begin{matrix}\left\{ \begin{matrix}{s = {r\theta}} \\{s \equiv \ {cabletranslatio{n\ ({mm})}}} \\{r \equiv \ {{pulley}\mspace{14mu}{{radius}\ ({mm})}}} \\{\theta \equiv \ {{rotation}\mspace{9mu}{of}\mspace{9mu}{{pully}\ ({rad})}}}\end{matrix} \right. & (15)\end{matrix}$

Such that the displacement, e.g., translation, of each cable 236, 246,256, 266 for movement in each of the DOFs is:

$\begin{matrix}\left\{ \begin{matrix}{s_{y} = {r_{y}\theta_{y}}} \\{s_{p} = {r_{p}\theta_{p}}} \\{s_{j}\  = {r_{j}\theta_{j}}}\end{matrix} \right. & (16)\end{matrix}$

As described above, the IDU 13 includes a motor (not shown) that isassociated with each of the motor interfaces 222, 224, 226, 228 torotate a respective one of the drive screws 230, 240, 250, 260 which inturn retract or relax a corresponding cable 236, 246, 256, 266. Themotors are rotated as a function of the desired pose of the end effector270 from the following:

$\begin{matrix}\left\{ \begin{matrix}{s = {\frac{k_{p}}{k_{m}}q}} \\{{k_{m} \equiv {{gear}\mspace{14mu}{ratio}}},\ {i.e.\mspace{9mu}\left( \frac{{input} - {motor}}{{output} - {motor}} \right)}\ ,\ {dimensionless}} \\{{k_{p} \equiv \ {{screw}\mspace{9mu}{pitch}}},\ {i.e.\mspace{9mu}\left( \frac{translation}{rotation} \right)}\ ,\ \left( \frac{mm}{rad} \right)} \\{q \equiv \ {{motor}\mspace{14mu}{{rotation}\ ({rad})}}}\end{matrix} \right. & (17)\end{matrix}$

Noting that s is the overall translation of the respective cable suchthat a desired rotation of a motor can be shown as a function of thedesired end effector pose as:

$\begin{matrix}{q_{n} = {\frac{k_{m}}{k_{p}}s_{n}}} & (18)\end{matrix}$

The desired rotation of each motor as a function of the desired endeffector pose can be combined in a single matrix as:

$\begin{matrix}{\begin{bmatrix}q_{1} \\q_{2} \\q_{3} \\q_{4}\end{bmatrix} = {{\frac{k_{m}}{k_{p}}\begin{bmatrix}{- r_{y}} & r_{p} & {0.5r_{p}} \\{- r_{y}} & {- r_{p}} & {{- 0.5}r_{p}} \\r_{y} & {- r_{p}} & {0.5r_{p}} \\r_{y} & r_{p} & {{- 0.5}r_{p}}\end{bmatrix}}\begin{bmatrix}\theta_{y} \\\theta_{p} \\\theta_{j}\end{bmatrix}}} & (19)\end{matrix}$

If it is assumed that radii of each of the idlers 273, 275 and spindles277, 279 are equal for each of the cables 236, 246, 256, 266, thenEquation (19) can be simplified to:

$\begin{matrix}{\begin{bmatrix}q_{1} \\q_{2} \\q_{3} \\q_{4}\end{bmatrix} = {{\frac{{rk}_{m}}{k_{p}}\begin{bmatrix}{- 1} & 1 & 0.5 \\{- 1} & {- 1} & {- 0.5} \\1 & {- 1} & 0.5 \\1 & 1 & {- 0.5}\end{bmatrix}}\begin{bmatrix}\theta_{y} \\\theta_{p} \\\theta_{j}\end{bmatrix}}} & (20)\end{matrix}$

From the inverse kinematic model of Equation (20), a forward kinematicmodel can be obtained to relate motor position to the yaw, pitch, andjaw angles of the end effector 270. It is noted that Equation (20) hasthree unknowns and four equations. However, if proper tension ismaintained in each of the cables 236, 246, 256, 266, Equation (20) canbe solved in the reverse direction. For example, to solve for θ_(j), thefirst and third equations or the second and fourth equations can beadded together. Since it is no more likely that either one of the sumsis more accurate, an average of the two can be taken. Following thisreasoning, a forward relationship can be expressed as follows:

$\begin{matrix}{\begin{bmatrix}\theta_{y} \\\theta_{p} \\\theta_{j}\end{bmatrix} = {{\frac{k_{p}}{4{rk}_{m}}\begin{bmatrix}{- 1} & {- 1} & 1 & 1 \\1 & {- 1} & {- 1} & 1 \\2 & {- 2} & 2 & {- 2}\end{bmatrix}}\begin{bmatrix}q_{1} \\q_{2} \\q_{3} \\q_{4}\end{bmatrix}}} & (21)\end{matrix}$

It will be appreciated that Equation (21) can be obtained by taking apseudoinverse, e.g., the Moore-Penrose inverse, of Equation (20).

Referring to FIG. 11, the motors of the IDU 13 can be controlled by aproportional-integral-derivative (PID) controller 300 as detailed belowin accordance with the present disclosure. The desired motor position{right arrow over (q_(d))} is calculated from a desired jaw angle {rightarrow over (θ_(d))} using Equation (19). The motor position is thensensed as {right arrow over (q_(s))} and subtracted from the desiredposition to generate a motor position error {right arrow over (q_(e))}.The PID controller 300 then calculates a motor torque {right arrow over(τ_(m))} which is outputted to the IDU 13. Propositional, integral, andderivative gains K_(p), K_(i), K_(d). can be expressed in a control lawas:

$\begin{matrix}{{\overset{\rightarrow}{\tau}}_{m} = {{K_{p}{\overset{\rightarrow}{q}}_{e}} + {K_{i}{\int_{t_{0}}^{t}{{\overset{\rightarrow}{q}}_{e}{dt}}}} + {K_{d}\frac{d}{dt}{\overset{\rightarrow}{q}}_{e}}}} & (22)\end{matrix}$

where {right arrow over (q_(e))} is calculated from Equation (20) asfollows:

$\begin{matrix}{{\overset{\rightarrow}{q}}_{e} = {{\overset{\rightarrow}{q}}_{d} - {{\frac{{rk}_{m}}{k_{p}}\begin{bmatrix}{- 1} & 1 & 0.5 \\{- 1} & {- 1} & {- 0.5} \\1 & {- 1} & 0.5 \\1 & 1 & {- 0.5}\end{bmatrix}}{\overset{\rightarrow}{q}}_{s}}}} & (23)\end{matrix}$

The PID controller 300 as shown in FIG. 11 may also be used to controlstiffness in one or more DOF of the end effector 270. In addition, thePID controller 300 may calculate error in joint space instead of motorspace as shown by utilizing the forward kinematics model of Equation(21).

As detailed above, the PID controller 300 alone does not directly orexplicitly account for force within the cables 236, 246, 256, 266. Forexample, maintaining tension, e.g., preventing slack in the cables, is arequirement of the kinematics models detailed above. It will beappreciated that if the motors of the IDU 13 track the desired motorposition with high precision and in sync with one another, that tensionwill be maintained in the cables 236, 246, 256, 266. In contrast, if themotors are unable to move to a desired motor position in sync with oneor more of the other motors, then one or more of the cables 236, 246,256, 266 may become slack. In embodiments, the cables 236, 246, 256, 266may be pre-tensioned by a predetermined amount by monitoring current ofthe motors of the IDU 13 and/or monitoring torques sensors before thePID controller 300. By pre-tensioning the cables 236, 246, 256, 266,tension in the cables will be kept positive even if the motors of theIDU 13 move the cables 236, 246, 256, 266 out of sync.

In addition, the PID controller 300 alone does not maintain a clampingforce in the jaws “a, b” when the jaws “a, b”, are closed. For example,when the motors of the IDU 13 close the position loop according toEquation (20), the net clamping force is approximately zero. Inembodiments, when the end effector 270 cannot reach a zero position,some closure force may remain when the jaws “a, b” are in the closedposition. In embodiments, the jaw angle can be commanded to be negativesuch that closure force may remain when the jaws “a, b” are in theclosed position. In such embodiments, a desired angle in the jaw DOF maybe scaled and the inverse kinematic equation may be modified once thejaw angle is past zero such that the clamping force is modulated bychanging a magnitude of the desired negative angle. However, thisapproach may cause confusion for a clinician as the end effector 270 maynot open when it is commanded until the commanded jaw angle exceeds thedesired negative angle. Further, there may be tracking errors in the jawDOF due to the rescaling.

In accordance with embodiments of the robotic surgical system 1 (FIG.1), the surgical robot 10 or processing unit 30 includes a null space(NS) controller 400 that is configured to provide cable tension and/orjaw clamping force. In such embodiments, the PID controller 300 is aprimary positional controller and the NS controller 400 is a secondarycontroller.

The PID controller 300 continues to control a pose of the end effector270 using Equation (20) as detailed above. As the PID controller 300controls the pose of the end effector 270, another DOF, cable force, iscreated in the motor space from Equation (13) which can be related tothe motor torque through the screws 230, 240, 250, 260. The cable forcein each cable 236, 246, 256, 266 is related to torque of the respectivemotor as:

f _(i) =k _(ls)τ_(i) ,i=1,2,3,4  (24)

where k_(ls) is a conversion coefficient that accounts for direction andefficiency of the respective screw 230, 240, 250, 260.

Assuming that the screws 230, 240, 250, 260 have the same radius,Equation (24) can be combined with Equation (13) such that:

$\begin{matrix}{\begin{bmatrix}\tau_{y} \\\tau_{p} \\\tau_{j}\end{bmatrix} = {{{rk}_{ls}\begin{bmatrix}\left( {1 - \eta_{2}} \right) & \left( {1 - \eta_{2}} \right) & {- \left( {1 + \eta_{2}} \right)} & {- \left( {1 + \eta_{2}} \right)} \\\left( {1 - \eta_{1a}} \right) & {- \left( {1 + \eta_{1a}} \right)} & {- \left( {1 + \eta_{1b}} \right)} & \left( {1 - \eta_{1b}} \right) \\{2\left( {1 - \eta_{1a}} \right)} & {{- 2}\left( {1 + \eta_{1a}} \right)} & {2\left( {1 + \eta_{1b}} \right)} & {{- 2}\left( {1 - \eta_{1b}} \right)}\end{bmatrix}}\begin{bmatrix}\tau_{1} \\\tau_{2} \\\tau_{3} \\\tau_{4}\end{bmatrix}}} & (25)\end{matrix}$

which can be simplified to:

$\begin{matrix}{S \equiv {{rk}_{ls}\begin{bmatrix}\left( {1 - \eta_{2}} \right) & \left( {1 - \eta_{2}} \right) & {- \left( {1 + \eta_{2}} \right)} & {- \left( {1 + \eta_{2}} \right)} \\\left( {1 - \eta_{1a}} \right) & {- \left( {1 + \eta_{1a}} \right)} & {- \left( {1 + \eta_{1b}} \right)} & \left( {1 - \eta_{1b}} \right) \\{2\left( {1 - \eta_{1a}} \right)} & {{- 2}\left( {1 + \eta_{1a}} \right)} & {2\left( {1 + \eta_{1b}} \right)} & {{- 2}\left( {1 - \eta_{1b}} \right)}\end{bmatrix}}} & (26)\end{matrix}$

As shown above, Equation 25 is under constrained such that for the samearticulating torque, there may be different ways to generate the motortorque to balance Equation (25). For example, motor torques {right arrowover (τ)}_(motor) may satisfy Equation (25) and the requestingarticulating joint torques {right arrow over (τ)}_(joint) such that:

{right arrow over (τ)}_(joint) =S{right arrow over (τ)}_(motor)  (27)

where the motor torques {right arrow over (τ)}_(motor) could be torquesthat are generated from the PID controller 300 as shown above withrespect to Equation (22). As Equation (25), and thus the matrix S ofEquation (26), is unconstrained, additional motor torques Δ{right arrowover (τ)}_(motor) may be added to the right side of Equation (27) suchthat:

{right arrow over (τ)}_(joint) =S({right arrow over (τ)}_(motor)+Δ{rightarrow over (τ)}_(motor))  (28)

This is satisfied by constraining the additional motor torques Δ{rightarrow over (τ)}_(motor) within null space of matrix S as:

{right arrow over (0)}=SΔ{right arrow over (τ)}_(motor)  (29)

such that there is no net change in the articulating joint torques{right arrow over (τ)}_(joint) determined by the PID controller 300.

From the above, it is possible to develop the secondary NS controller400 to adjust the motor torques {right arrow over (τ)}_(motor) generatedby the PID controller 300 such that the NS controller 400 cascades afterthe PID controller 300 to directly adjust the motor torques {right arrowover (τ)}_(motor). The motor torques {right arrow over (τ)}_(motor) maybe measured directly by additional torque sensors that detect the outputtorque of the motors of the DU 13. For a detailed description ofsuitable torque sensors, reference can be made to U.S. Pat. No.9,987,094, the entire contents of which are hereby incorporated byreference.

First, maintaining tension in the cables 236, 246, 256, 266 will beaddressed in accordance with the present disclosure. During normaloperation the end effector 270 has three DOF with the articulatingtorques defined in Equation (13). These torques can be adjusted withoutimpacting the position of the end effector 270 as long as the additionalmotor torques Δ{right arrow over (τ)}_(motor) satisfy Equation (29) suchthat vectors in null space can be expressed as:

Δ{right arrow over (τ)}_(motor)=(I _(4×4) −S ^(T)(SS ^(T)))⁻¹ S){rightarrow over (z)}  (30)

where matrix I is a 4×4 identity matrix; S^(T) is the transpose ofmatrix S, and {right arrow over (z)} is an arbitrary vector with adimension of 4. The arbitrary vector {right arrow over (z)} can beprojected or decomposed through a helper matrix by:

N=(I _(4×4) −S ^(T)(SS ^(T))⁻¹ S)  (31)

Then, substituting Equation (26) into Equation (31) and setting thefriction coefficients μ₁ and μ₂ to zero produces:

$\begin{matrix}{N = \begin{bmatrix}0.25 & 0.25 & 0.25 & 0.25 \\0.25 & 0.25 & 0.25 & 0.25 \\0.25 & 0.25 & 0.25 & 0.25 \\0.25 & 0.25 & 0.25 & 0.25\end{bmatrix}} & (32)\end{matrix}$

Next, the influence of the constant rK_(ls) is extracted from the nullspace defined in Equation (29) by assuming that the arbitrary vector{right arrow over (z)} has the same unit of measure as the motor torques{right arrow over (τ)}_(motor). By substituting Equation (32) intoEquation (30):

$\begin{matrix}{{\Delta{\overset{\rightarrow}{\tau}}_{motor}} = {\begin{bmatrix}0.25 & 0.25 & 0.25 & 0.25 \\0.25 & 0.25 & 0.25 & 0.25 \\0.25 & 0.25 & 0.25 & 0.25 \\0.25 & 0.25 & 0.25 & 0.25\end{bmatrix}\begin{bmatrix}z_{1} \\z_{2} \\z_{3} \\z_{4}\end{bmatrix}}} & (33)\end{matrix}$

which is rearranged as:

$\begin{matrix}{\begin{bmatrix}{\Delta\tau}_{{motor},1} \\{\Delta\tau}_{{motor},2} \\{\Delta\tau}_{{motor},3} \\{\Delta\tau}_{{motor},4}\end{bmatrix} = {0.25\begin{bmatrix}{z_{1} + z_{2} + z_{3} + z_{4}} \\{z_{1} + z_{2} + z_{3} + z_{4}} \\{z_{1} + z_{2} + z_{3} + z_{4}} \\{z_{1} + z_{2} + z_{3} + z_{4}}\end{bmatrix}}} & (34)\end{matrix}$

where the additional motor torque vector Δ{right arrow over (τ)}_(motor)is explicitly expressed by its elements as:

$\begin{matrix}{{\Delta{\overset{\rightarrow}{\tau}}_{motor}} = \begin{bmatrix}{\Delta\tau}_{{motor},1} \\{\Delta\tau}_{{motor},2} \\{\Delta\tau}_{{motor},3} \\{\Delta\tau}_{{motor},4}\end{bmatrix}} & (35)\end{matrix}$

and since arbitrary vector {right arrow over (z)} is an arbitraryvector, another arbitrary scalar Δτ can be defined as:

Δτ=0.25(z ₁ +z ₂ +z ₃ +z ₄)  (36)

which results in:

$\begin{matrix}{\begin{bmatrix}{\Delta\tau}_{{motor},1} \\{\Delta\tau}_{{motor},2} \\{\Delta\tau}_{{motor},3} \\{\Delta\tau}_{{motor},4}\end{bmatrix} = \begin{bmatrix}{\Delta\tau} \\{\Delta\tau} \\{\Delta\tau} \\{\Delta\tau}\end{bmatrix}} & (37)\end{matrix}$

As shown by inserting Equation (37) into Equation (28), it is clear thatthe same torque can be added to each motor of the IDU 13 withoutchanging the articulating torques for the yaw, pitch, and jaw DOFs.Thus, if the additional motor torque for each motor is the same, aposition determined by the PID controller 300 will not be impacted. Asdetailed above with respect to Equation (25) this assumes that thefriction is ignored which is allowed as the additional torque of eachmotor does not articulate the joints but generates internal forceswithin the system and contributes to a stiffness of the system.

However, when the friction coefficients μ₁ and μ₂ are significant, thematrix S is solved symbolically such that the null space of Equation(29) is expressed as:

$\begin{matrix}{{\Delta{\overset{\rightarrow}{\tau}}_{motor}} = {\begin{bmatrix}\frac{\left( {1 + \eta_{1a}} \right)\left( {1 + \eta_{2}} \right)}{\left( {1 + \eta_{1b}} \right)\left( {1 + \eta_{2}} \right)} \\\frac{\left( {1 - \eta_{1a}} \right)\left( {1 + \eta_{2}} \right)}{\left( {1 + \eta_{1b}} \right)\left( {1 - \eta_{2}} \right)} \\{\frac{1 - \eta_{1b}}{1 + \eta_{1b}}\mspace{101mu}} \\{1\mspace{169mu}}\end{bmatrix}{\Delta\tau}}} & (38)\end{matrix}$

It is noted that if the friction coefficients μ₁ and μ₂ are zero,Equation (38) degrades to Equation (37) such that Equation (37) is afirst order approximation of Equation (38). Thus, Equation (38) providesrelationships of the additional motor torques Δ{right arrow over(τ)}_(motor) that can be adjusted without impacting the articulatingtorques in jaw, pitch, and jaw DOFs to avoid impacting the positiondetermined by the PID controller 300.

As Equation (38) is flexible in the implementation of the NS controller400, several methods may be used to adjust the torque for each motor ofthe IDU in view of a measured torque {right arrow over (τ)}_(s).

In a first method of maintaining cable tension, torque is adjusted ateach motor according to a measured torque {right arrow over (τ)}_(s)which can be taken from a measured current or from a physical torquesensor as detailed above. In practice, when the NS controller 400 isoperating in the first method, positional gain of the PID controller 300as shown in Equation (22) is necessarily high. Further, to maintainminimum tension f_(min) in the cables 236, 246, 256, 266, two of thefour motors of the IDU 13 keep minimum tensions in the respective cableswhile the other two motors of the IDU 13 take over a workload from thetwo motors maintaining the minimum tension f_(min) in addition to theexpected workload. This is reflected in Table 1, above, which requiresthe positional change be distributed to the motor pairs with differentsigns by the same amounts such that each pair of motors of the IDU 13that are physical connected by a pair of cables, e.g., first and secondcables 236, 246 and third and fourth cables 256, 266.

The differential drive mechanism detailed above is a push-pull mechanismwith two pairs of motors. However, in the first method the pusher doesnot actively push but maintains the minimum tension and the puller motorpulls the pusher motor to the desired position such that the pullermotor of each pair closes the positional loop for both motors of thepair which requires the proportional gains to be high and the maximumcurrent of the puller motor to be increased significantly. Even in viewof the increased gains, the first method is extremely good atmaintaining the minimum tension f_(min) in each cable.

To keep the tension of each cable 236, 246, 256, 266 at a minimumtension f_(min) as the motors of the IDU 13 articulate the end effector270, a minimum Δτ is determined as:

$\begin{matrix}{\begin{bmatrix}{{\frac{\left( {1 + \eta_{1a}} \right)\left( {1 + \eta_{2}} \right)}{\left( {1 + \eta_{1b}} \right)\left( {1 - \eta_{2}} \right)}{\Delta\tau}} + \tau_{s,1}} \\{{\frac{\left( {1 - \eta_{1a}} \right)\left( {1 + \eta_{2}} \right)}{\left( {1 + \eta_{1b}} \right)\left( {1 - \eta_{2}} \right)}{\Delta\tau}} + \tau_{s,2}} \\{{{\frac{1 - \eta_{1b}}{1 + \eta_{1b}}{\Delta\tau}} + \tau_{s,3}}\mspace{101mu}} \\{{{\Delta\tau} + \tau_{s,4}}}\end{bmatrix} \geq {\frac{1}{k_{ls}}\begin{bmatrix}f_{\min} \\f_{\min} \\f_{\min} \\f_{\min}\end{bmatrix}}} & (39)\end{matrix}$

If the adjustment torque Δτ is the minimum torque that satisfiesEquation (29), the adjustment torque Δτ from the NS controller 400 isexpressed as:

$\begin{matrix}{{\overset{\rightarrow}{\tau}}_{null} = \begin{bmatrix}{\frac{\left( {1 + \eta_{1a}} \right)\left( {1 + \eta_{2}} \right)}{\left( {1 + \eta_{1b}} \right)\left( {1 - \eta_{2}} \right)}{\Delta\tau}} \\{\frac{\left( {1 - \eta_{1a}} \right)\left( {1 + \eta_{2}} \right)}{\left( {1 + \eta_{1b}} \right)\left( {1 - \eta_{2}} \right)}{\Delta\tau}} \\{{\frac{1 - \eta_{1b}}{1 + \eta_{1b}}{\Delta\tau}}\mspace{101mu}} \\{{\Delta\tau}}\end{bmatrix}} & (40)\end{matrix}$

In this first method, torque is adjusted at each individual motor of theIDU 13. The measured torque {right arrow over (τ)}_(s) is checked andadjusted to ensure that the tension in each of the cables 236, 246, 256,266 is greater than the minimum tension f_(min).

In a second method of maintaining cable tension, the physical connectionbetween each pair of cables, e.g., first and second cable 236, 246 andthird and fourth cable 256, 266, are considered such that theconstraints defined in Equation (39) are relaxed. Specifically, insteadof verifying torque of each motor, the torque of each pair of motors ofthe IDU 13 is constrained by an average of each motor group or pair ofmotors of the IDU 13.

If the average of the measured torque {right arrow over (τ_(s))} meetsthe requirements as detailed below, then the puller motor of each pairis working harder than the pusher motor but since there is still a nettorque in each of the motors of the pair, e.g., the pusher and puller,it means that the cable of the pusher motor is also likely undertension. However, as each motor is not individually checked, the secondmethod may not provide as consistent result in maintaining tension ineach of the cables greater than the minimum tension f_(min). However, asthe puller motor of each pair of motors is not required to take theentire load of the pusher motor of each pair of motors such that theload on the motors of the IDU 13, and the stiffness of the joint, can bereduced.

In the second method, the average of each of the motor groups isexpressed as:

$\begin{matrix}{\begin{bmatrix}{{\frac{\left( {1 + \eta_{1a}} \right)\left( {1 + \eta_{2}} \right)}{\left( {1 + \eta_{1b}} \right)\left( {1 - \eta_{2}} \right)}\Delta\tau} + \tau_{s,1} + {\frac{\left( {1 - \eta_{1a}} \right)\left( {1 + \eta_{2}} \right)}{\left( {1 + \eta_{1b}} \right)\left( {1 - \eta_{2}} \right)}\Delta\tau} + \tau_{s,2}} \\{{\frac{1 - \eta_{1b}}{1 + \eta_{1b}}\Delta\tau} + \tau_{s,3} + {\Delta\tau} + \tau_{s,4}}\end{bmatrix} \geq {\frac{1}{2k_{ls}}\begin{bmatrix}f_{\min} \\f_{\min}\end{bmatrix}}} & (41)\end{matrix}$

Similar to the first method, the minimum adjustment torque Δτ isdetermined that fulfills Equation (41) such that output from the NScontroller 400 takes the form of Equation (40). It is not necessary toshow that if Equation (39) holds that Equation (41) will also hold.However, it is also clear that Equation (41) provides a less certainconstraint that the minimum tension f_(min) of each cable is maintained.However, as detailed above, as a result of each pair of cables beingcoupled together, as long as the puller motor of each pair of motors hasa measured torque {right arrow over (τ_(s))} greater than the average,then it is likely that the pusher motor is also maintaining the minimumtension f_(min) in its related cable. Further, it is noted that Equation(41) accounts for an influence of friction in the calculation of theadjustment torque Δτ.

In a third method of maintaining cable tension, the measured torque{right arrow over (τ_(s))} of each motor of the IDU 13 is tracked andmaintained above a targeted minimum torque τ_(min) based on a measuredtorque {right arrow over (τ_(s))} that corresponds to the minimumtension f_(min) desired in each of the cables 236, 246, 256, 266. Thethird method has a simple controller in the form of:

Δτ=τ_(min) +K(™_(in)−τ_(s,min))  (42)

where τ_(min) is the targeted a minimum torque; τ_(s, min) is a sensedminimum torque, and K is a gain factor.

The adjustment torque Δτ in Equation (42) can provide varying degrees ofcertainty of maintaining tension in the cables 236, 246, 256, 266. Forexample, when the gain factor K=1, the adjustment torque Δτ will adjustevery motor torque {right arrow over (τ)}_(motor) by the minimum torqueτ_(s), min such that each motor torque {right arrow over (τ)}_(motor)will be at or above the targeted minimum torque τ_(min) to maintain theminimum tension f_(min) in the cables 236, 246, 256, 266. This will havea similar result as the first method detailed above where the pullermotor of each pair of motors will take on the entire workload of therespective pusher motor while providing certainty that the minimumtension f_(min) is maintained. When the gain factor K<1, certainty ofmaintaining the minimum tension f_(min) is reduced while the additionalworkload on the puller motor of each pair of motors is reduced. In anextreme case, when the gain factor K=0, a constant is added to eachmotor torque {right arrow over (τ)}_(motor) to maintain the minimumtension f_(min).

The third method allows for consideration of friction from Equation (38)by scaling the four elements of Equation (38) to account for frictionand to make the highest adjustment torque Δτ equal to the output ofEquation (42).

The third method allows for allowing for tuning of the gain factor K toaccount for performance requirements of other controllers such as anoverall power, a maximum current that the motors of the IDU 13 canprovide, and/or a stiffness of the joint. This allows for the gainfactor K to be increased when certainty of maintaining cable tension iscritical and workload on the motors of the IDU 13 can be increased andto be reduced when certainty of maintaining cable tension is lesscritical and workload on the motors of the IDU 13 needs to be reduced.

Second, generating a clamping force between jaws “a, b” will beaddressed in accordance with the present disclosure. It may bebeneficial to adjust the clamping force between jaws “a, b” depending onthe type of instrument of the end effector 270. For example, when thejaws “a, b” form a needle driver, a high clamping force is required andwhen the jaws “a, b” are a bowel grasper a much lower clamping force isrequired.

In a first method or technique of generating clamping force, theclamping force is generated by over clamping the jaws, e.g., commandingthe desired jaw angle {right arrow over (θ_(d))} to a negative value. Inthe first technique, an amount of clamping force generated depends onthe overall stiffness of the PID controller 300 and a magnitude of thedesired negative angle. To maintain cable tension, Equation (20) ismodified after the actual jaw angle passes zero as follows:

$\begin{matrix}{\begin{bmatrix}q_{1} \\q_{2} \\q_{3} \\q_{4}\end{bmatrix} = {{\frac{rk_{m}}{k_{p}}\begin{bmatrix}{- 1} & 1 & 0 \\{- 1} & {- 1} & {- {0.5}} \\1 & {- 1} & 0 \\1 & 1 & {- {0.5}}\end{bmatrix}}\begin{bmatrix}\theta_{y} \\\theta_{p} \\\theta_{j}\end{bmatrix}}} & (43)\end{matrix}$

By modifying Equation (20), the puller motors of each pair of motorscontinue to pull while the pusher motors are prevented from relaxing.

As a commanded jaw angle θ_(j) does not go negative, the commanded jawangle θ_(j) must be remapped in the first technique. As detailed above,if the commanded jaw angle θ_(j) is remapped linearly, a clinician maynotice a significant difference between a commanded jaw angle θ_(j) andan actual jaw angle. To improve a clinician's experience and moreclosely track the commanded jaw angle θ_(j) with the actual jaw angle,the commanded jaw angle θ_(j) can be mapped to the actual jaw anglenonlinearly. The commanded jaw angle θ_(j) can be mapped using differentslopes where for most of a range of the commanded jaw angle θ_(j) theslope is substantially linear and as the commanded jaw angle θ_(j)approaches zero, a steeper slope is used. For example, a logarithmicfunction may be used such that when the commanded jaw angle θ_(j)∈[0,10]can be mapped to [−10, 10] roughly through 6 log₄(θ_(j)+0.1). When thecommanded jaw angle θ_(j) changes from 0.9 to 10, the mapped angle θ_(m)changes from 0 to 10 such that the mapping is substantially linear. Incontrast, when the commanded jaw angle θ_(j) changes from 0 to 0.9, themapped angle θ_(m) changes from −10 to 0 such that the slope issignificantly non-linear and steep.

In a second method or technique for generating a clamping force betweenjaws “a, b”, the NS controller 400 can be used to generate the clampingforce when the jaws “a, b” are in closed position. When the jaws “a, b”are in the closed position, the end effector 270 has two DOF such thatEquation (11) degrades, as the articulating torque τ_(j) of in the jawDOF is now zero, to the following:

$\begin{matrix}{\quad\left\{ \begin{matrix}{\tau_{y} \equiv {{\left( {f_{1} + f_{2}} \right)r_{y}} - {\left( {f_{4} + f_{3}} \right)r_{y}} - {{\mu_{2}\left( {f_{1} + f_{2} + f_{3} + f_{4}} \right)}{{sgn}\left( {\overset{.}{\theta}}_{y} \right)}r_{y}}}} \\\begin{matrix}{\tau_{p} \equiv {{\left( {f_{1} - f_{2}} \right)r_{p}} - {{\mu_{1}\left( {f_{1} + f_{2}} \right)}{{sgn}\left( {\overset{.}{\theta}}_{p} \right)}r_{p}} +}} \\{{\left( {f_{4} - f_{3}} \right)r_{p}} - {{\mu_{1}\left( {f_{3} + f_{4}} \right)}{{sgn}\left( {\overset{.}{\theta}}_{p} \right)}r_{p}}}\end{matrix}\end{matrix} \right.} & (44)\end{matrix}$

Two new variables can be defined as follows:

$\begin{matrix}\left\{ \begin{matrix}{\eta_{1} = {\mu_{1}{{sgn}\left( {\overset{.}{\theta}}_{p} \right)}}} \\{\eta_{2} = {\mu_{2}{{sgn}\left( {\overset{.}{\theta}}_{y} \right)}}}\end{matrix} \right. & (45)\end{matrix}$

to simplify Equation (44) to:

$\begin{matrix}{\begin{bmatrix}\tau_{y} \\\tau_{p}\end{bmatrix} = {r{{k_{ls}\begin{bmatrix}\left( {1 - \eta_{2}} \right) & \left( {1 - \eta_{2}} \right) & {- \left( {1 + \eta_{2}} \right)} & {- \left( {1 + \eta_{2}} \right)} \\\left( {1 - \eta_{1}} \right) & {- \left( {1 + \eta_{1}} \right)} & {- \left( {1 + \eta_{1}} \right)} & \left( {1 - \eta_{1}} \right)\end{bmatrix}}\begin{bmatrix}\tau_{1} \\\tau_{2} \\\tau_{3} \\\tau_{4}\end{bmatrix}}}} & (46)\end{matrix}$

Similarly, Equation (26) also degrades to:

$\begin{matrix}{S = {r{k_{ls}\begin{bmatrix}\left( {1 - \eta_{2}} \right) & \left( {1 - \eta_{2}} \right) & {- \left( {1 + \eta_{2}} \right)} & {- \left( {1 + \eta_{2}} \right)} \\\left( {1 - \eta_{1}} \right) & {- \left( {1 + \eta_{1}} \right)} & {- \left( {1 + \eta_{1}} \right)} & \left( {1 - \eta_{1}} \right)\end{bmatrix}}}} & (47)\end{matrix}$

which can be expressed for null space of matrixes S_(i) to S₄ as definedin Equation (47) as:

$\begin{matrix}{{\Delta{\overset{\rightarrow}{\tau}}_{motor}} = {{\begin{bmatrix}\frac{1 + \eta_{1}}{1 - \eta_{2}} \\\frac{\eta_{2} - \eta_{1}}{1 - \eta_{2}} \\1 \\0\end{bmatrix}\Delta\tau_{3}} + {\begin{bmatrix}\frac{\eta_{2} - \eta_{1}}{1 - \eta_{2}} \\\frac{1 + \eta_{1}}{1 - \eta_{2}} \\0 \\1\end{bmatrix}\;{\Delta\tau}_{4}}}} & (48)\end{matrix}$

where Δτ₃ and Δτ₄ are arbitrary scalars.

By setting the friction coefficients μ₁ and μ₂ to zero it can be shownthat the additional motor torques Δ{right arrow over (τ)}_(motor) formotors 2 and 4 are equal and that the additional motor torques Δ{rightarrow over (τ)}_(motor) for motors 1 and 3 are equal:

$\begin{matrix}{\begin{bmatrix}{\Delta\tau_{{motor},1}} \\{\Delta\tau_{{motor},2}} \\{\Delta\tau_{{motor},3}} \\{\Delta\tau_{{motor},4}}\end{bmatrix} = \begin{bmatrix}{\Delta\tau_{3}} \\{\Delta\tau_{4}} \\{\Delta\tau_{3}} \\{\Delta\tau_{4}}\end{bmatrix}} & (49)\end{matrix}$

Thus, to balance internal forces in the end effector 270:

f _(clamp) =f ₄ −f ₃ =f ₂ −f ₁  (50)

which constrains the additional motor torques Δ{right arrow over(τ)}_(motor) to:

f _(clamp) =k _(ls)Δτ_(clamp) =f ₂ −f ₁  (51)

f _(clamp) =k _(ls)Δτ_(clamp) =k _(ls)(Δτ_(motor,4)−Δτ_(motor,3))=k_(ls)(Δτ_(motor,2)−Δτ_(motor,1))  (52)

Equation (52) illustrates that the difference between Δτ₃ and Δτ₄ can beused to adjust the clamping force Δτ_(clamp) between jaws “a, b” whichis defined as:

Δτ_(clamp)=Δτ₄−Δτ₃  (53)

Such that Equation (49) can be rewritten as:

$\begin{matrix}{\begin{bmatrix}{\Delta\tau_{{motor},1}} \\{\Delta\tau_{{motor},2}} \\{\Delta\tau_{{motor},3}} \\{\Delta\tau_{{motor},4}}\end{bmatrix} = \begin{bmatrix}{\Delta\tau_{3}} \\{{\Delta\tau_{3}} + {\Delta\tau_{clamp}}} \\{\Delta\tau_{3}} \\{{\Delta\tau_{3}} + {\Delta\tau_{clamp}}}\end{bmatrix}} & (54)\end{matrix}$

When Equation (54) is compared to Equation (37) it is clear that cabletension can be maintained in the first, second, and third methods whileadjusting the clamping force Δτ_(clamp) between jaws “a, b” once thejaws “a, b” are in the closed position.

When the friction coefficients μ₁ and μ₂ are included, the clampingforce Δτ_(clamp) of Equation (50) depends on the sign in Equation (45).For example, when sgn(Op) is positive, the second part of Equation (44)is rearranged as:

τ_(p) =r _(p){(f ₄ −f ₃)−μ₁(f ₃ +f ₄)−((f ₂ −f ₁)+μ₁(f ₁ +f ₂))}  (55)

In this example, the articulating torque for pitch τ_(p) is provided bythe fourth cable 266. To provide the articulating torque for pitchτ_(p), the frictional forces that must be overcome by the fourth cable266 are proportional to the total force of the third and fourth cables256, 266 and represented by μ₁(f₃+f₄). In addition, the fourth cable 266must also overcome a countering force provided by the opposite jaw whichis shown as f₂−f₁. Further, the fourth cable 266 must also overcome thefriction of the other jaw represented as μ₁(f₁+f₂). The internal torqueor clamping force Δτ_(clamp) can be expressed as:

Δτ_(clamp) =r _(p)(f ₂ −f ₁+μ₁(f ₁ +f ₂))  (56)

Further, the motor torques Δ{right arrow over (τ)}_(motor) can absorbedthe coefficient k_(ls) as:

Δτ_(clamp)=Δτ_(motor,2)−Δτ_(motor,1)+μ₁(Δτ_(motor,1)+Δτ_(motor,2))  (57)

To maintain the clamping force Δτ_(clamp) between jaws “a, b”, Equation(48) can be expressed as:

$\begin{matrix}{{\Delta{\overset{\rightarrow}{\tau}}_{motor}} = \begin{bmatrix}{{\frac{1 + \mu_{1}}{1 - \eta_{2}}\Delta\tau_{3}} + {\frac{\eta_{2} + \mu_{1}}{1 - \eta_{2}}\Delta\tau_{4}}} \\{{\frac{\eta_{2} - \mu_{1}}{1 - \eta_{2}}\Delta\tau_{3}} + {\frac{1 - \mu_{1}}{1 - \eta_{2}}\Delta\tau_{4}}} \\{\Delta\tau_{3}} \\{\Delta\tau_{4}}\end{bmatrix}} & (58)\end{matrix}$

Substituting Equation (57) into Equation (58) in which Δτ₄ is expressedas a function of Δτ₃ and the clamping force Δτ_(clamp) yields:

$\begin{matrix}{{\Delta{\overset{\rightarrow}{\tau}}_{motor}} = \begin{bmatrix}\frac{{\left( {\mu_{1} + \eta_{2}} \right)\Delta\tau_{clamp}} + {\left( {1 + \eta_{2}} \right)\left( {1 + \mu_{1}} \right)\Delta\tau_{3}}}{\left( {1 - \eta_{2}} \right)\left( {1 - \mu_{1}} \right)} \\\frac{{\Delta\tau_{clamp}} + {\left( {1 + \eta_{2}} \right)\Delta\tau_{3}}}{1 - \eta_{2}} \\{\Delta\tau_{3}} \\\frac{{\Delta\tau_{clamp}} + {\left( {1 + \mu_{1}} \right)\Delta\tau_{3}}}{1 - \mu_{1}}\end{bmatrix}} & (59)\end{matrix}$

In a similar manner, when sgn({dot over (θ)}_(p)) is negative, thesecond part of Equation (44) can be rearranges as:

τ_(p) =−r _(p){(f ₂ −f ₁)−μ₁(f ₁ +f ₂)−((f ₄ −f ₃)+μ₁(f ₃ +f ₄))}  (60)

In this example, the articulating torque for pitch is provided by thesecond cable 246 such that the internal torque or clamping forceΔτ_(clamp) can be expressed as:

Δτ_(clamp)=Δτ_(motor,4)−Δτ_(motor,3)+μ₁(Δτ_(motor,3)+Δτ_(motor,4))  (61)

To maintain the clamping force Δτ_(clamp) between jaws “a, b”, Equation(48) can be expressed as:

$\begin{matrix}{{\Delta{\overset{\rightarrow}{\tau}}_{motor}} = \begin{bmatrix}{{\frac{1 - \mu_{1}}{1 - \eta_{2}}\Delta\tau_{3}} + {\frac{\eta_{2} - \mu_{1}}{1 - \eta_{2}}\Delta\tau_{4}}} \\{{\frac{\eta_{2} + \mu_{1}}{1 - \eta_{2}}\Delta\tau_{3}} + {\frac{1 + \mu_{1}}{1 - \eta_{2}}\Delta\tau_{4}}} \\{\Delta\tau_{3}} \\{\Delta\tau_{4}}\end{bmatrix}} & (62)\end{matrix}$

Substituting Equation (61) into Equation (61) provides:

$\begin{matrix}{{\Delta{\overset{\rightarrow}{\tau}}_{motor}} = \begin{bmatrix}\frac{{\left( {{- \mu_{1}} + \eta_{2}} \right)\Delta\tau_{clamp}} + {\left( {1 + \eta_{2}} \right)\left( {1 - \mu_{1}} \right)\Delta\tau_{3}}}{\left( {1 - \eta_{2}} \right)\left( {1 + \mu_{1}} \right)} \\\frac{{\Delta\tau_{clamp}} + {\left( {1 + \eta_{2}} \right)\Delta\tau_{3}}}{1 - \eta_{2}} \\{\Delta\tau_{3}} \\\frac{{\Delta\tau_{clamp}} + {\left( {1 - \mu_{1}} \right)\Delta\tau_{3}}}{1 + \mu_{1}}\end{bmatrix}} & (63)\end{matrix}$

A governing equation can be developed in view of the symmetry betweenEquations (59) and (63) as:

$\begin{matrix}{\left\lbrack \begin{matrix}{\Delta\tau_{{motor},1}} \\{\Delta\tau_{{motor},2}} \\{\Delta\tau_{{motor},3}} \\{\Delta\tau_{{motor},4}}\end{matrix} \right\} = \begin{bmatrix}\frac{{\left( {\eta_{1} + \eta_{2}} \right)\Delta\tau_{clamp}} + {\left( {1 + \eta_{2}} \right)\left( {1 + \eta_{1}} \right)\Delta\tau_{3}}}{\left( {1 - \eta_{2}} \right)\left( {1 - \eta_{1}} \right)} \\\frac{{\Delta\tau_{clamp}} + {\left( {1 + \eta_{2}} \right)\Delta\tau_{3}}}{1 - \eta_{2}} \\{\Delta\tau_{3}} \\\frac{{\Delta\tau_{clamp}} + {\left( {1 + \eta_{1}} \right)\Delta\tau_{3}}}{1 - \eta_{1}}\end{bmatrix}} & (64)\end{matrix}$

which has two independent variables, the clamping force Δτ_(clamp) andΔτ₃. The clamping force Δτ_(clamp) can be used to control an amount offorce between jaws “a, b” such that Equation (64) provides a directmeans for managing a clamping torque and force in the jaw DOF. Further,the free variable Δτ₃ can be used to maintain cable tension as detailedabove.

Equation (64) depends on the assumption that the jaws “a, b” of the endeffector 270 are closed such that the DOF of the end effector 270 arereduced to two such that Equation (44) holds. In use, this condition canbe evaluated through the forward kinematic model of Equation (21). Toverify the forward kinematic model of Equation (21) it may beadvantageous to use the motor positions provided by encoders for themotors of the IDU 13 due to the physical separation of the motors andthe end effector 270. As the motor encoders will induce someuncertainty, a clamping threshold can be set such that once thecalculated jaw angle is smaller than the clamping threshold, theclamping torque adjustment as noted in Equation (64) can be appliedgradually over a short period of time, e.g., 0.5 seconds. Further, asthe adjustments are proportional to the two free variables, the clampingforce Δτ_(clamp), and Δτ₃, if the adjustments are increasedproportionally, the position controlled by the PID controller 300 may beimpacted.

As the clamping force is only provided on instruction from a clinicianinterfacing with the robotic surgical system 1 (FIG. 1), when aclinician signals an intent to open the jaws “a, b” of the end effector270 the clamping force may be released faster than the addition ofclamping force, e.g., 0.1 seconds. As the clinician can visually observethe jaws “a, b” opening once the intent is clear, a desired jaw angle{right arrow over (θ_(d))} can be used to determine a releasingthreshold instead of a calculated jaw angle as detailed above. Once thedesired jaw angle {right arrow over (θ_(d))} crosses the clampingthreshold, the clamping torque in the form of the clamping forceΔτ_(clamp) can be quickly applied. In addition, a hysteresis can bebuilt between the clamping threshold and the releasing threshold toavoid a constant crossing between applying a clamping force andreleasing a clamping force.

Referring now to FIG. 12, a total controller 600 that combines the PIDcontroller 300, the NS controller 400, and a combination controller 500provided in accordance with the present disclosure to control theposition of the end effector 270 in yaw, pitch, and jaw and to maintaincable tension and generate a clamping force when needed. The PIDcontroller 300 receives the desired jaw angle {right arrow over (θ_(d))}and outputs motor torques {right arrow over (τ)}_(m) to the NScontroller 400. In addition, the PID controller 300 receives theposition of the motors {right arrow over (q)}_(s) of the IDU 13 as partof a feedback loop.

The NS controller 400 receives the motor torques {right arrow over(τ)}_(m) from the PID controller 300 and the desired jaw angle {rightarrow over (θ_(d))}. The NS controller 400 can utilize any of themethods for maintaining cable tension and/or techniques for generating aclamping force as detailed above. The desired jaw angle {right arrowover (θ_(d))} can be used to calculate a desired angular velocity abouteach joint and to pick a corresponding sign for the frictioncoefficients μ₁ and μ₂, e.g., opposite the direction of desired angularvelocity. Alternatively, friction may be ignored to simplify the NScontroller 400. The NS controller 400 also receives the position of themotors {right arrow over (q)}_(s) of the IDU 13 as part of a feedbackloop and uses the forward kinematics model to generate calculated jointangles {right arrow over (θ_(s))} to detect when the system loses a DOF,e.g., when the calculated joint angles {right arrow over (θ_(s))} areless than a clamping threshold, to determine when to apply a clampingtorque. In addition, the NS controller 400 may also receive a sensedtorque {right arrow over (τ)}_(s) to maintain cable tension.

The NS controller 400 outputs the motor torques {right arrow over(τ)}_(m) and null torques {right arrow over (τ)}_(null) to thecombination controller 500. The combination controller 500 adds the nulltorques {right arrow over (τ)}_(null) to the motor torques {right arrowover (τ)}_(m) to calculate a desired torque {right arrow over (τ)}_(d)from a sum of the motor torques {right arrow over (τ)}_(m) and nulltorques {right arrow over (τ)}_(null) for output to the IDU 13. Inembodiments where a sensed torque {right arrow over (τ)}_(s) isavailable, the combination controller 500 receives the sensed torque{right arrow over (τ)}_(s) and closes the desired combination controller500 then outputs the desired torque {right arrow over (τ)}_(d). Theproportional gain K_(s) is implemented to reduce error between thedesired torque {right arrow over (τ)}_(d) and the sensed torque {rightarrow over (τ)}_(s). Specifically, a sum of the motor torques {rightarrow over (τ)}_(m) and null torques {right arrow over (τ)}_(null) maybe the desired torque {right arrow over (τ)}_(d) and the proportionalgain K_(s) is implemented such that the sensed torque {right arrow over(τ)}_(s) approaches the sum of the motor torques {right arrow over(τ)}_(m) and null torques {right arrow over (τ)}_(null). Thus, when anerror between the desired torque {right arrow over (τ)}_(d) and thesensed torque {right arrow over (τ)}_(s) is small, the proportional gainK_(s) is also small. As detailed herein, the combination controller 500is a tertiary controller.

Referring to FIG. 13, another total controller 610 that combines the PIDcontroller 300 and the NS controller 400 is disclosed in accordance withthe present disclosure to control the position of the end effector 270in yaw, pitch, and jaw and to maintain cable tension. As shown, the NScontroller 400 adds the motor torques {right arrow over (τ)}_(m) and thenull torques {right arrow over (τ)}_(null) and outputs the desiredtorque {right arrow over (τ)}_(d) directly to the IDU 13. The totalcontroller 610 may be used when the NS controller 400 maintains cabletension but does not generate a clamping force.

With reference to FIG. 14, another total controller 620 that combinesthe PID controller 300 and the NS controller 400 is disclosed inaccordance with the present disclosure to control the position of theend effector 270 in yaw, pitch, and jaw and to maintain cable tensionand generate a clamping force when needed. The total controller 620 alsoincludes a joint space converter 622 that converts the motor positions{right arrow over (q)}_(s) of the IDU 13 from the motor space to jointangles {right arrow over (θ_(s))} in the joint space by using theforward kinematics model of Equation (21). The joint space converter 622delivers the joint angles {right arrow over (θ_(s))} to the PIDcontroller 300 and the NS controller 400. The total controller 620 mayalso include a distribution controller 310 as detailed below which alsoreceives the joint angles {right arrow over (θ_(s))} from the jointspace controller 622. The distribution controller 310 may be a separatecontroller or may be integrated into the PID controller 300.

By operating the PID controller 300 directly in the joint space, theyaw, pitch, and jaw joints can have different controlled stiffnesses ifdesired. The output from the PID controller 300 in the joint space isjoint torque {right arrow over (τ)}_(j) which is calculate as:

$\begin{matrix}{{\overset{\rightarrow}{\tau}}_{j} = {{K_{p}{\overset{\rightarrow}{\theta}}_{e}} + {K_{i}{\int_{t_{0}}^{t}{{\overset{\rightarrow}{\theta}}_{e}dt}}} + {K_{d}\frac{d}{dt}{\overset{\rightarrow}{\theta}}_{e}}}} & (65)\end{matrix}$

The distribution controller 310 receives the joint torques {right arrowover (τ)}_(j) from the PID controller 300 and distributes the jointtorques {right arrow over (τ)}_(j) to the motor torques {right arrowover (τ)}_(m). However, there is not an equation that directlydistributes the joint torques {right arrow over (τ)}_(j) to motortorques {right arrow over (τ)}_(m). Equation (25) relates motor torques{right arrow over (τ)}_(m) to joint torques {right arrow over (τ)}_(j)but is not invertible. Similar to the forward kinematics, apseudoinverse of Equation (26) can be used to distribute the jointtorques {right arrow over (τ)}_(j) to motor torques {right arrow over(τ)}_(m). The distribution controller 310 may account for friction whendistributing the joint torques {right arrow over (τ)}_(j) to the motortorques {right arrow over (τ)}_(m). The distribution controller 310 thenoutputs the motor torques {right arrow over (τ)}_(m) to the NScontroller 400. As noted above, the distribution controller 310 may beseparate from or integrated into the PID controller 300.

The NS controller 400 then receives the motor torques {right arrow over(τ)}_(m) from the distribution controller 310 and calculates the nulltorques {right arrow over (τ)}_(null), adds the motor torques {rightarrow over (τ)}_(m) and the null torques {right arrow over (τ)}_(null),and outputs the desired torque {right arrow over (τ)}_(d) directly tothe IDU 13 in a manner similar to those detailed above.

The overall controllers 600, 610, 620 are examples of contemplatedoverall controllers and should not be seen as limiting. Other overallcontrollers are also considered to accommodate the differential drivemechanism which controllers maintain cable tension and generate clampingforces using a null space controller. For example, another overallcontroller may verify the desired torque {right arrow over (τ)}_(d)before the desired torque {right arrow over (τ)}_(d) is delivered to theIDU 13 to verify that the desired torque {right arrow over (τ)}_(d) iswithin an acceptable range for the IDU 13. In such a controller, if oneor more of the desired torques {right arrow over (τ)}_(d) is outside ofa range for the IDU 13, then a null space technique may be used toadjust the desired torques {right arrow over (τ)}_(d) before outputtingthe desired torques {right arrow over (τ)}_(d) to the IDU 13, e.g., thecable tension may be reduced and/or the clamping force may be reduced.

While several embodiments of the disclosure have been shown in thedrawings, it is not intended that the disclosure be limited thereto, asit is intended that the disclosure be as broad in scope as the art willallow and that the specification be read likewise. Any combination ofthe above embodiments is also envisioned and is within the scope of theappended claims. Therefore, the above description should not beconstrued as limiting, but merely as exemplifications of particularembodiments. Those skilled in the art will envision other modificationswithin the scope of the claims appended hereto.

What is claimed:
 1. An adapter for a surgical tool, the surgical tooldefining a longitudinal axis, the adapter comprising: a first drivescrew longitudinally fixed and configured to rotate about a first screwaxis parallel to the longitudinal axis, the first drive screw having athreaded portion; a first drive nut disposed about the threaded portionof the first drive screw and threadably coupled to the first drive screwsuch that the first drive nut longitudinally translates in response torotation of the first drive screw and the first drive screw rotates inresponse to longitudinal translation of the first drive nut; a firstcable having a proximal portion fixed to the first drive nut and adistal portion; a first spring disposed about the first drive screw andconfigured to urge the first drive nut in a first longitudinaldirection, the first spring having a first spring constant; a seconddrive screw longitudinally fixed and configured to rotate about a secondscrew axis parallel to first screw axis, the second drive screw having athreaded portion; a second drive nut disposed about the threaded portionof the second drive screw and threadably coupled to the second drivescrew such that the second drive nut longitudinally translates inresponse to rotation of the second drive screw and the second drivescrew rotates in response to longitudinal translation of the seconddrive nut; a second cable having a proximal portion fixed to the seconddrive nut and a distal portion, the distal portions of the first andsecond cables operatively coupled to one another such that translationsof the distal portions oppose one another; and a second spring disposedabout the second drive screw and configured to urge the second drive nutin the first direction, the second spring having a second springconstant, the second spring biased such that the second springtranslates the second drive nut and second cable in the first directionsuch that the second cable translating the first cable and the firstdrive nut in a second direction opposite the first direction and againstthe bias of the first spring such that the tool is biased towards apredetermined pose.
 2. The adapter according to claim 1, wherein thefirst screw includes a first proximal head configured to interface witha first motor and the second screw includes a second proximal headconfigured to interface with a second motor.
 3. The adapter according toclaim 1, wherein the first direction is proximal and the seconddirection is distal.
 4. The adapter according to claim 1, wherein thefirst drive nut defines a first slot, the proximal portion of the firstcable fixed in the first slot.
 5. The adapter according to claim 1,wherein the second spring constant is larger than the first springconstant.
 6. A surgical tool comprising: an elongate shaft defining alongitudinal axis, the elongate shaft having a proximal end and a distalend; an end effector supported adjacent the distal end of the elongateshaft and including a first jaw and a second jaw movable in pitch, yaw,and jaw DOFs; and an adapter supporting the proximal end of the elongateshaft, the adapter including: a first drive screw longitudinally fixedwithin the adapter and configured to rotate about a first screw axisparallel to the longitudinal axis, the first drive screw having athreaded portion; a first drive nut disposed about the threaded portionof the first drive screw and threadably coupled to the first drive screwsuch that the first drive nut longitudinally translates in response torotation of the first drive screw and the first drive screw rotates inresponse to longitudinal translation of the first drive nut; a firstcable extending through the elongate shaft and having a proximal portionfixed to the first drive nut and a distal portion secured to the endeffector; a first spring disposed about the first drive screw andconfigured to urge the first drive nut in a first longitudinaldirection, the first spring having a first spring constant; a seconddrive screw longitudinally fixed within the adapter and configured torotate about a second screw axis parallel to first screw axis, thesecond drive screw having a threaded portion; a second drive nutdisposed about the threaded portion of the second drive screw andthreadably coupled to the second drive screw such that the second drivenut longitudinally translates in response to rotation of the seconddrive screw and the second drive screw rotates in response tolongitudinal translation of the second drive nut; a second cableextending through the elongate shaft and having a proximal portion fixedto the second drive nut and a distal portion secured to the endeffector, the distal portions of the first and second cables operativelycoupled to one another such that translations the distal portions opposeone another; and a second spring disposed about the second drive screwand configured to urge the second drive nut in the first direction, thesecond spring having a second spring constant, the second spring biasedsuch that the second spring translates the second drive nut and secondcable in the first direction such that the second cable translates thefirst cable and the first drive nut in a second direction opposite thefirst direction and against the bias of the first spring such that theend effector is biased towards a predetermined pose.
 7. The toolaccording to claim 6, wherein the distal portions of the first andsecond cables are each coupled to the first jaw.
 8. The tool accordingto claim 6, wherein the adapter includes: a third drive screwlongitudinally fixed within the adapter and configured to rotate about athird screw axis parallel to the longitudinal axis, the third drivescrew having a threaded portion; a third drive nut disposed about thethreaded portion of the third drive screw and threadably coupled to thethird drive screw such that the third drive nut longitudinallytranslates in response to rotation of the third drive screw and thethird drive screw rotates in response to longitudinal translation of thethird drive nut; a third cable extending through the elongate shaft andhaving a proximal portion fixed to the third drive nut and a distalportion secured to the end effector; a third spring disposed about thethird drive screw and configured to urge the third drive nut in a thirdlongitudinal direction, the third spring having a third spring constant;a fourth drive screw longitudinally fixed within the adapter andconfigured to rotate about a fourth screw axis parallel to third screwaxis, the fourth drive screw having a threaded portion; a fourth drivenut disposed about the threaded portion of the fourth drive screw andthreadably coupled to the fourth drive screw such that the fourth drivenut longitudinally translates in response to rotation of the fourthdrive screw and the fourth drive screw rotates in response tolongitudinal translation of the fourth drive nut; a fourth cableextending through the elongate shaft and having a proximal portion fixedto the fourth drive nut and a distal portion secured to the endeffector, the distal portions of the third and fourth cables operativelycoupled to one another such that translations of the distal portionsoppose one another; and a fourth spring disposed about the fourth drivescrew and configured to urge the fourth drive nut in the firstdirection, the fourth spring having a fourth spring constant, the fourthspring biased such that the fourth spring translates the fourth drivenut and fourth cable in the first direction, the fourth cabletranslating the third cable and the third drive nut in the seconddirection and against the bias of the third spring such that the endeffector is biased towards the predetermined pose.
 9. The tool accordingto claim 8, wherein the distal portion of the first cable is secured toa first side of the first j aw, the distal portion of the second cableis secured to a second side of the first jaw, the distal portion of thethird cable is secured to the second side of the second jaw, and thedistal portion of the fourth cable is secured to the first side of thesecond jaw such that the first and fourth cables are disposed on thesame side of the first and second jaws, respectively, and the second andthird cables are disposed on the same side of the first and second jaws,respectively.
 10. The tool according to claim 9, wherein the endeffector includes a yoke and a clevis, the clevis fixed to the distalend of the elongate shaft, the yoke pivotally coupled to the clevisabout a first axis perpendicular to and intersected by the longitudinalaxis, and the jaws pivotally coupled to the yoke about a second axisperpendicular to the first axis.
 11. The tool according to claim 10,wherein the first jaw has a first spindle pivotal about the second axisand the second jaw has a second spindle pivotal about the second axis.12. The tool according to claim 11, wherein the distal portions of thefirst and second cables are secured to opposite sides of the firstspindle and the distal portions of the third and fourth cables aresecured to opposite sides of the second spindle.
 13. The tool accordingto claim 10, wherein the second and fourth springs are configured tomaintain the tool in a pose with the first and second jaws in a closedposition, the first and second jaws longitudinally aligned with thelongitudinal axis, and the yoke aligned with the longitudinal axis. 14.The tool according to claim 8, wherein the first, second, third, andfourth cables are configured to manipulate the pose of the end effectorin pitch, yaw, and jaw DOFs.
 15. A surgical tool configured toselectively connect to a drive unit, the tool comprising: an elongateshaft defining a longitudinal axis, the elongate shaft having a proximalend and a distal end; an end effector supported adjacent the distal endof the elongate shaft and including a first jaw and a second jaw; afirst cable extending through the elongate shaft and having a distalportion secured to a first side of the first jaw; a second cableextending through the elongate shaft and having a distal portion securedto a second opposite side of the first jaw; a third cable extendingthrough the elongate shaft and having a distal portion secured to thesecond side of the second jaw; a fourth cable extending through theelongate shaft and having a distal portion secured to the first side ofthe second jaw; and an adapter supporting the proximal end of theelongate shaft and configured to selectively connect to a drive unit,the adapter including a differential drive mechanism configured tomanipulate proximal portions of each of the first, second, third, andfourth cables to manipulate the end effector in pitch, yaw, and jawDOFs, each of the first, second, third, and fourth cables biasedproximally and configured to maintain the end effector in a desired posewhen the tool is disconnected from a drive unit.
 16. The tool accordingto claim 15, wherein the adapter urges each of the first and thirdcables proximally with a first force and urges each of the second andfourth cables proximally with a second force greater than the firstforce.
 17. The tool according to claim 16, wherein the desired pose isstraight in pitch and yaw with the first and second jaws in a closedposition such that the first and second jaws are closed and aligned withthe longitudinal axis.
 18. The tool according to claim 15, wherein thedesired pose is straight in pitch and yaw with the first and second jawsin an open position such that the first and second jaws are spaced apartfrom one another and aligned with the longitudinal axis.
 19. The toolaccording to claim 15, wherein the end effector includes a yoke and aclevis, the clevis fixed to the distal end of the elongate shaft, theyoke pivotally coupled to the clevis about a first axis perpendicular toand intersected by the longitudinal axis, and the jaws pivotally coupledto the yoke about a second axis perpendicular to the first axis.
 20. Thetool according to claim 19, wherein the first jaw has a first spindlepivotal about the second axis and the second jaw has a second spindlepivotal about the second axis.